Quant Quiz

1. Sonu invested 10% more than Mona. Mona invested 10% less than Raghu. If the total sum of their investment is Rs.5780, how much amount did Raghu invest? 

(a)    Rs. 2010

(b)   Rs. 2100

(c)    Rs. 2000

(d)   Rs. 2200

(e)   None Of these

2. Three men and 5 women together can do a job in 3 days. Working on the same job 3 women take 5 days more than the time required by 2 men . What is the ratio of efficiency of a man to a woman?

3. In an examination, 44% of students failed in subject A and 36% of the students failed in subject B and 18% failed in both the subjects. If 352 students passed in only one subject, then how many failed in both the subjects?

(a)    126

(b)   144

(c)    156

(d)   158

(e)   None of these

4. A shopkeeper allows two successive discounts of 10% and 5% on the marked price of an article but charges a sales tax of 5% on the discounted price. If a customer pays Rs. 718.2 as including the sales tax, then what is the marked price?

(a)    750

(b)   780

(c)    800

(d)   830

(e)   None of these

5. Ram is 33(1/3)% as efficient as shyam. Gangu does 40% of the work done by Ram and Shyam together. If Gangu alone does the work in 60 days, in how many days Shyam and Gangu together can finish the same work?

(a)    20(2/3) days

(b)   25(4/5) days

(c)    20(10/23) days

(d)   Can not be determined

(e)   None of these.

6. The ratio of present ages of A and C is 7 : 9 and present age of B is equal to the average ages of A and C after 2 years. If the ratio of present age of B and age of C after four years is 6 : 7 then what will be the ratio of age of A after four years to that of present age of B?

(a)    13:14

(b)   14:13

(c)    6:7

(d)   7:6

(e)   None of these

7. A certain amount of money gets four times to its orignal amount after 24 years at a simple interest. What will be the interest on a sum of Rs.8500 compounded annually at the same rate of interest after 2 year?

(a)    2220.50

(b)   2150.75

(c)    2400.50

(d)   2257.60

(e)   None of these

8. A shopkeeper mixes of 24 kg of sugar worth Rs. 35 per kg with another quality of sugar of Rs.42 per kg. By selling the mixture at Rs. 41.04 per kg, he may gain 8%. How much sugar of second quality has been added to the mixture?

(a)    18 kg

(b)   32 kg

(c)    16 kg

(d)   28 kg

(e)   None of these

Reasoning

Directions (Q. 27-31): In each of the given questions, some statements are followed by two conclusions I and II. You have to assume everything in the statement to be true even if they seem to be at variance with commonly known facts, and then decide which of the given conclusions logically follows from the statements, disregarding commonly known facts.Give answer:
1) if only conclusion I follows.
2) if only conclusion II follows.
3) if either I or II follows.
4) if neither I nor II follows.
5) if both conclusions I and II follow

27. Statements : Some resources are powerful., All resources are significant , All significant are organised.
Conclusions: I. Some organised are not significant.
                      II. All organised being powerful is a possibility.

28. Statements: All regular are successful, Some funds are regular, All exercises are funds.
Conclusions: I. All exercises being successful is a possibility.
                      II. At least some regular are funds.

29. Statements: All regular are successful, Some funds are regular, All exercises are funds.
Conclusions: I. Some successful being exercises is a possibility.
                      II. Some funds can never be exercises.

30. Statements : No society is standard, No standard is modern, No modern is official.
Conclusions: I. Some societies being modern is a possibility.
                      II. Some societies can never be official.

31. Statements : All attire are balanced. All balanced are critical. Some critical are delightful. All delightful are effective.
Conclusions: I. All critical being attire is a possibility.
                      II. Some balanced are delightful.

Directions (Q. 32-36): Study the following information carefully and answer the questions given below.
A, B, C, D, E and F are six ambassadors of six different countries viz. Brazil, India, China, France, Australia, Russia but not necessarily in the same order. They are sitting around a circular table for discussing six different topics viz Trade, LOC, Terrorism, World peace, Environment and Development. But not necessarily in the same order. Only one person gives his views on one topic only.
(a) ‘Development, Terrorism and World Peace’ pointing persons are neither from ‘France’ nor from ‘Brazil’.
(b) Ambassadors from ‘India’ and ‘China’ are neither is pointing ‘Development’ nor ‘Terrorism’.
(c) A is neither from ‘France’ nor on the immediate left of the person who has pointed ‘Environment’.
(d) The only ambassador who is between ‘E’ and ‘F’ is pointing ‘LOC’.
(e) ‘D’ has his discussion about ‘Environment’ and he is from ‘China’. He is facing the person who has his views on ‘LOC’.
(f) The one who is delivering about ‘Development’ is seated opposite to the person who is from ‘Russia’, while China’s ambassador is on the left of the person, who is noticing World Peace.
(g) One who discussed about ‘Trade’ is on the immediate right of France’s ambassador but on the immediate left of the person who has briefed about ‘Terrorism’.
(h) ‘C’ is not pointing ‘Terrorism’ while ‘F’ has no points for ‘Development’. F has discussion about trade and immediate left of the person who dicussion terrorism.

32. Who among the following is France's ambassador?
1) A         2) B        3) C        4) E         5) None of these

33. The only ambassador between E and D is noticing which of the following topics?
1) World Peace     2) LOC      3) Development     4) Environment        5) None of these

34. Who among the following has a discussion about 'World Peace'?
1) E         2) B         3) A        4) C         5) E

35. Which of the following is correctly matched?
1) A-India-Environment        2) B-Australia-Terrorism        3) E-Australia-Development
4) F-Brazil-World Peace       5) None of these

36. 'Trade' pointing person is from which country?
1) Australia         2) India        3) Russia       4) Brazil        5) None of these

27. 2
28. 5
29. 1
29. 1
31. 1

32-36) 

32- 3           33- 1           34-3           35-3           36-4

Data Interpretation- Caselets

Directions: (1-5) Study the following information and answer the questions that follow: (IBPS CWE PO MT 2012)
The premises of a bank are to be renovated. The renovation is in terms of flooring. Certain areas are to be floored either with marble or wood. All rooms/halls and pantry are rectangular. The area to be renovated comprises of a hall for customer transaction measuring 23 m by 29 m, branch manager’s room measuring 13 m by 17 m, a pantry measuring 14 m by 13 m, a record keeping cum server room measuring 21 m by 13 m and locker area measuring 29 m by 21 m. The total area of the bank is 2000 square meters. The cost of wooden flooring is f 170/- per square meter and the cost of marble flooring is Rs. 190/- per square meter. The locker area, record keeping cum server room and pantry are to be floored with marble. The branch manager's room and the hall for customer transaction are to be floored with wood. No other area is to be renovated in terms of flooring.

1. What is the respective ratio of the total cost of wooden flooring to the total cost of marble flooring? 
(1) 1879: 2527
(2) 1887: 2386
(3) 1887: 2527
(4) 1829: 2527
(5) 1887: 2351

2. If the four walls and ceiling of the branch manager’s room (The height of the room is 12 meters) are to be painted at the cost off 190/- per square meter, how much will be the total cost of renovation of the branch manager's room including the cost of flooring?
(1) Rs. 1, 36,800/-
(2) Rs. 2, 16,660/-
(3) Rs. 1, 78,790/-
(4) Rs. 2, 11,940/-
(5) None of these

3. If the remaining area of the bank is to be carpeted at the rate of Rs. 110/- per square meter, how much will be the increment in the total cost of renovation of bank premises? 
(1) Rs. 5,820/-
(2) Rs. 4,848/-
(3) Rs. 3,689/-
(4) Rs. 6,890/-
(5) None of these

4. What is the percentage area of the bank that is not to be renovated? 
(1) 2.2 %
(2) 2.4 %
(3) 4.2 %
(4) 4.4 %
(5) None of these

5. What is the total cost of renovation of the hall for customer transaction and the locker area? 
(1) Rs. 2, 29,100
(2) Rs. 2, 30,206
(3) Rs. 2, 16,920
(4) Rs. 2, 42,440
(5) None of these

Directions (Q. 6-10): Study the following information carefully to answer the questions that follow: (RBI Grade’B’ Officer’s Exam 2011) 
There are two trains, Train A and Train B. Both trains have four different types of coaches, viz General, Sleeper, First Class and AC. In Train A, there are total 700 passengers. Train B has thirty per cent more passengers than Train A. Twenty per cent of the passengers of Train A are in General Coach. One-fourth of the total number of passengers of Train A are in AC coach. Twenty three per cent of the passengers of Train A are in Sleeper Coach. Remaining passengers of Train A are in First Class Coach. The total number of passengers in AC Coach in both the trains together is 480. Thirty per cent of the number of passengers of Train B are in Sleeper Coach. Ten per cent of the total passengers of Train B are in First Class Coach. The remaining passengers of Train B are in General Coach.

6. What is the ratio of the number of passengers in First Class Coach of Train A to the number of passengers in Sleeper Coach of Train B?
(1) 13 : 7
(2) 7 : 13
(3) 32 : 39
(4) Data Inadequate
(5) None of these

7. What is the total number of passengers in the General Coach of Train A and the AC Coach of Train B together?
(1) 449
(2) 459
(3) 435
(4) 445
(5) None of these

8. What is the difference between the number of passengers in the AC Coach of Train A and the total number of passengers in Sleeper and First Class Coach together of Train B?
(1) 199
(2) 178
(3) 187
(4) 179
(5) None of these

9. The total number of passengers in General Coaches of both the trains together is approximately what percentage of the total number of passengers in Train B?
(1) 35
(2) 42
(3) 46
(4) 38
(5) 31

10. If the cost per ticket of First Class coach is Rs.450, what will be the total amount generated from First Class Coach of Train A?
(1) Rs.1, 00, 080
(2) Rs.1, 08, 000
(3) Rs.1, 00, 800
(4) Rs.10, 800
(5) None of these

1.  (3) 1887: 2527

Solution: Total flooring area with marble = locker area + record keeping + pantry

= 182+273 +609 = 1064 sqm

 Cost of flooring = 1064*190

Total flooring area with wooden = Branch Manager room + Hall

= 221 + 667 = 888 sqm

Cost of flooring = 888*170

Ratio= 888*170: 1064*190 = 888*17: 1064*19 = 15096 : 20216 = 1887: 2527

2. (5) None of these

Solution: Cost of flooring of branch manager room =221*170= Rs. 37570

Cost of painting = [2(17*12+13*12)+ 13*17]*190

= [2( 204 + 156) + 221]*190 = (2*360 + 221)*190

= (720 + 221)*190 = 941*190 = Rs. 178790

Total cost = 178790 + 31570 = Rs.216360

3. (5) None of these

Solution: Total area of bank = 2000 sqm

Total flooring area = 1952 sqm

Remaining area = 2000 - 1952 = 48 sqm

Cost of carpeting = 48 x 110 = Rs.5280

4. (2) 2.4%

Solution:- Area not to be renovated = 48 sqm

Therefore, Required % = 48*100/2000 = 2.4%

5. (1) Rs. 2,29,100

Solution: Cost of renovation of hall + locker area

= 667x170 + 609x190

= 113390 + 115710 = Rs. 229100

6. 3            7. 4            8. 5            9. 2            10. 3

Reasoning Quiz

Directions (Q. 1-5): Study the following information and answer the following questions :
M, N, O, P, Q, R and T seven friends working in different companies - HCL, Ranbaxy, Wipro, DLF, Reliance, Tata and Samsung. Each of them has a different car viz Wagonr, Brio, Santro, Alto, Indica, Etios and Swift, but not necessarily in the same order. R works in Wipro and drives santro car. T drives swift car and does not works in Samsung. M works in Tata and does not drive either wagonr or Brio. O drives Alto car and does not work either in Reliance or Ranbaxy. P drives Indica car and works in DLF. Q does not work in Ranbaxy. N does not drive Etios. One who drives Wagnor works in Reliance.

1. Which of the following car does Q drives ?
1) Etios     2) Brio     3) Wagonr     4) Swift     5) None of these

2. In which company does O works ?
1) Reliance    2) Ranbaxy    3) Samsung     4) Can't be determined     5) None of these

3. In which company does N works ?
1) DLF     2) Ranbaxy     3) Samsung     4) HCL     5) None of these

4. Who drives Brio car ?
1) M      2) Q     3) T     4) N     5) P

5. Which of the following combination is true ?
1) T - Swift - Samsung     2) N - Wagonr - Reliance      3) N - Wagonr - Ranbaxy
4) O - Alto - Samsung      5) None of these


Directions (Q. 6- 10) : In each of the questions given below are two/three statements followed by two conclusions numbered I and II. You have to take the given statement to be true even if they seem to be at Variance with commonly known facts and then decide which of the given conclusions logically follows from the given statements, disregarding commonly known facts. Give answer :
1) If only conclusion I follows.
2) If only conclusion II follows.
3) If either conclusion I or conclusion II follows.
4) If neither conclusion I nor conclusion II follows.
5) If both conclusion I and II follow.

(Q. 6 -7) : Statements :- All chairs are papers. No paper is a coin. All coin are statues.

6. Conclusion:- I. No statue is a chair. II. All papers being statues is a possibility.

7. Conclusion:- I. No chair is a coin. II. All chairs being statue is a possibility.

8. Statements :- No window is door. some frames are doors. No metal is frame.
    Conclusion:- I. No window is a metal. II. No door is a metal.

9. Statements :- All books are machines. All machines are pens. Some pens are erasers.
    Conclusion:- I. It is a possibility that all erasers are books.
                           II. At least some Pens are books.

10. Statements :- All stones are birds. All birds are beautiful.
      Conclusion:- I. All beautiful things being stones is a possibility.
                           II. Some birds are not stones.

Directions (Q. 11 - 15) : Study the following information carefully and answer the given questions :

A word arrangement machine when given an input line of words and numbers rearranges them following a particular rule the following is an illustration of input and rearrangement.

Input :19 out 10 full 18 13 lust evil 5 point 15 appear 23 ice.
Step - I : appear 5 19 out 10 full 18 13 lust evil point 15 23 ice.
Step -II: appear 5 evil 13 19 out 10 full 18 lust point 15 23 ice.
Step -III: appear 5 evil 13 ice 19 out 10 full 18 lust point 15 23.
Step -IV: appear 5 evil 13 ice 19 out 23 10 full 18 lust point 15.
Step -V: appear 5 evil 13 ice 19 out 23 point 10 full 18 lust 15.
Step -VI: appear 5 evil 13 ice 19 out 23 point 10 lust 15 full 18.

Step VI is the last step of the rearrangement. As per rules followed in above steps, find out, in each of the following questions, the appropriate steps for the input given below.
Input: east 17 altitude 31 united 39 food 23 14 in nest 33 come 11 rest.

11. How many steps will be needed to complete the rearrangement ?
1) Four     2) Five     3) Six     4) Seven     5) Eight

12. Which step would be the following output ?
      'altitude 11 east 17 in 23 united 31 39 food 14 nest 33 come rest'.
1) Step II    2) Step III    3) Step IV    4) Step V    5) None of these

13. Which of the following would be at 4th position from the right in step V ?
1) 14         2) 39       3) food     4) 33     5) None of these

14. Which of the following represent the position of 'rest' in step IV ?
1) Seventh from right       2) Eight from left       3) First from right
4) Fifth from right            5) None of these

15. Which element is fourth to the left of 'food' in step II ?
1) 31         2) 23            3) 17          4) east          5) united

Data Interpretation for SBI Mains

Direction(1-5):- Following table shows the production and sale (in thousands) of tyres of six different companies A, B, C, D, E and F during the period 2008 to 2013.

1. In which year is the number of tyres sold for Company E the minimum of its tyres produced?
a) 2008
b) 2009
c) 2010
d) 2011
e) 2012

2. What is the difference between the total tyres produced in all the six companies in 2012 and the total tyres sold in 2013(answer in thousands)?
a) 40
b) 41
c) 42
d) 43
e) 44

3. What is the percentage increase/decrease in the number of tyres sold by Company A from 2011 to 2012?
a) 12%
b) 12.5%
c) 13%
d) 13.5%
e) None of these

4. What is the ratio of total tyres sold by Company A and E together in the year 2008 to that in the year 2013?
a) 6:5
b) 5:6
c) 7:6
d) 6:7
e) None of these

5. In Company A, the number of tyres sold in 2012 is what percent more than the number of tyres sold in the year 2008?
a) 10%
b) 15%
c) 20%
d) 25%
e) 30%

Direction(6-10): Following line graph shows the ratio of expenditure to income of three companies A,B and C during the period 2008-2013.

6. In which of the following years is the percentage loss/gain pf company C the maximum?
1) 2008      2) 2009      3) 2010      4) 2011      5) 2012

7. If the expenditure of Company A in 2008 and 2009 together is Rs 60 lakhs, then what is its income in 2008 and 2009 together?
1)  Rs 120 lakhs   2) Rs 150 lakhs   3) Rs 66.66 lakhs   4) Data Inadequate   5) None of these

8. If the expenditure of Company B in 2008 and 2012 together is Rs 60 lakhs then what is its income in 2008 and 2012 together?
1) Rs 66.66 lakhs  2) Rs 75 lakhs    3) Rs 48 lakhs   4) 96 Rs lakhs   5) Data inadequate

9. In which of the years does Company C gain 100% profit?
1) 2008        2) 2009       3) 2010       4) 2011        5) None of these

10. What is the percentage decrease in the percentage profit of Company C from 2009 to 2010?
1) 75%        2) 300%        3) 62.5%       4) 160%      5) None of these

Data Interpretation

Directions (Q. 1-5): Read the table carefully to answer the questions that follows :
Percentage wise share distribution is given of different companies P, Q, R, S and T out of the total shares.

1. Total shares sold by companies P, R and T in year 2001 is what percentage of the total share of company R ?
1) 24.67           2) 21.67           3) 25.67             4) 28.67           5) None of these

2. What is the average share sold by all the companies in year 2004 ?
1) 175              2) 160              3) 165               4) 155               5) None of these

3. What is the difference of share sold by companies P, T and Q in year 2005 together and same in year 2004 ?
1) 205              2) 204               3) 206              4) 305                5) None of these

4. What is the average share sold by company R in all the years together ?
1) 298              2) 200               3) 198              4) 398                5) None of these

5. Total no. of shares sold by companies P, Q and S in 2003 together ?
1) 493.5           2) 393.5            3) 593.5          4) 429                 5) None of these


Direction (Q. 6-10): Read the line graph carefully to answer the questions below :

6. What is the total number of students hired by TCS for all centres ?
1) 2150          2) 2350                 3) 2450        4) 3000               5) None of these

7. Total no. of students heired by all the companies for Indore centre is ?
1) 2500          2) 2400                 3) 2450        4) 2300               5) None of these

8. What is the difference between total number of students heired by all the companies for lucknow centre and Indore centre ?
1) 950            2) 850                   3) 900          4) 925                 5) None of these

9. What is the average no. of students heired by TCS for all centres taken together ?
1) 530            2) 630                   3) 430          4) 420                 5) None of these

10.What is the difference between total no. of students hired by all companies for Indore and average no. of students for the same ?
1) 1825          2) 1925                 3) 1725       4) 2025                5) None of these

Answers :)

10. 3; 2300 - 2300/4 = 1725

Permutation And Combination

                 PERMUTATIONS AND COMBINATIONS

Factorial Notation

The continued product of first n natural numbers is called n factorial or factorial n.
It is denoted by | n or n!

Thus, | n or n! = 1. 2. 3. 4. .... (n -1) . n
                       = n (n -1)(n -2). .... 3.2.1          (in reverse order)

Meaning of zero factorial

According to the above definition, makes no sense. However we define 0! = 1

Note. When n is a negative integer or a fraction, n factorial is not defined.

Permutations and Combinations

Combination

Each of the groups or selections which can be made by taking some or all of a number of things without reference to the order of things in each group is called acombination.
For example, choosing two fruits out of a basket containing 4 oranges and 5 bananas - three combinations are possible - two oranges, two bananas or one orange and one banana.

Permutation

Each of the arrangements which can be made by taking some or all of a number of things is called permutation.
For example, the letters of word CAT can be arranged in six different ways
     CAT, CTA, ACT, ATC, TAC, TCA

Sometimes, students may be confused whether combination or permutation is being talked about. Certain key words/phrases can be helpful in deciding.
Combinations: Selection, choose, make group, distribute, committee, geometric problems.
Permutations: Arrangements, standing in a line, seated around a table, problems on digits or letters of a word.

Fundamental Theorem

1. Multiplication Principle (Principle of Association).

If a certain thing can be done in m ways, and if when it has been done, a second thing can be done in n ways, then the total number of ways in which two things can be done is mn.
For example, if there are eight top heroes and 4 top heroines, a film producer can choose from 8 x 4 = 32 pairs.

2. Addition Principle:

If there are two jobs which can be done in m and n ways respectively, then either of two jobs can be done in m + n ways.
For example, if the film producer has limited budget and he can afford either a top hero or a top heroine but not both, then he can choose in 8 +4 = 12 ways.

Example

Three persons enter a railway carriage, where there are 5 vacant seats. In how many ways can they seat themselves?

Solution

First man can sit on any of 5 vacant seats. Then the second can sit on any of 4 vacant seats left. And the third can sit on any of 3 vacant seats left. Hence by fundamental principle of counting, the required number of ways is 5 x 4 x 3 = 60.

Formula for P(n,r)

The number of permutations of n different things taken r at a time is given by

P(n, r) = n. (n -1). (n -2) ... (n -r +1) = n! / (n -r)!

Note. ⁿ P_n = n (n -1) ... (n - n +1) = n (n -1) ... 1 = n!

Example

^1)  5P₅ = 5. 4. 3. 2. 1 = 120

2) In how many ways can the letters of the word WONDERFUL be arranged?

    SOLUTION: 

                Since there are 9 distinct letters in WONDERFUL, the required number of permutations is

                ⁹P₉ = 9! = 9. 8. 7. 6. 5. 4. 3. 2. 1 = 362880

3) Find the number of divisors of the number 36000.

    Solution: Factorising the given number, we find 36000 = 2⁵ . 3². 5³. This means that any divisor of 36000 is of the type 2^a. 3^b. 5^c where a can take values 0, 1, 2, 3, 4, 5; b can take values 0, 1, 2; c can take values 0, 1,2, 3. Hence number of divisors is 6 x 3 x 4 = 72. Note that both 1 and 36000 are counted among 72 divisors.

Permutations under restrictions

• Number of permutations of n distinct objects when a particular object is not taken in any arrangement is ^n-1P_r
• Number of permutations of n distinct objects when a particular object is always included in any arrangement is r.^n-1P_r-1.

Example

In how many ways can 4 books on Mathematics and 3 books on English be placed on a shelf so that books on the same subject always remain together?

Solution

Consider the 4 books on Mathematics as one big book and 3 books on English as another big book. These two can be arranged in 2! ways. In each of these arrangements, 4 books on Mathematics can be arranged among themselves in 4! ways and 3 books on English can be arranged among themselves in 3! ways.
Hence, the required number of ways = 4! 3! 2! = 288.

Permutations of objects which are not all different

• The number of permutations of n things taken all together, when p of the things are alike of one kind, q of them alike of another kind, r of them alike of a third kind and the remaining all different is n! / [p! q! r!]
• The number of permutations of n objects, of which m are of one kind and the rest n -m of another kind, taken all at a time is n! / [m! (n-m)!]

Example

In how many ways can the letters of the word permutations be arranged such that

1. there is no restriction
2. P comes before S
3. words start with P and end with S
4. T's are together
5. vowels are together
6. order of vowels remains unchanged?

Solution

The given word has 12 letters -two Ts and 10 different letters.

1. Total number of arrangements is12!/2! = 6. 11!
2. Out of above, P comes before S in half the arrangements. Hence required number of arrangements = 3. 11!
3. As position of P and S is fixed, remaining 10 letters (that is, two T's and eight other different letters) can be arranged in 10!/2! = 5. 9! ways.
4. Considering two T's as a block, we have to arrange 11 different things, which can be done in 11! ways.
5. Considering the five vowels in given letter -E, U, A, I, O as a block, we have 8 things having 2 alike things (T's). So this can be arranged in 8!/2! = 4. 7! ways. Now within the block, 5 different vowels can be arranged in 5! ways. Hence required number of arrangements is 4. 7!5! ways.
6. If order of five vowels has to remain unchanged, we can consider them like five alike things, so only one ordering is possible. Thus we have 12 things of which 2 are alike and 5 are alike. Hence required number of arrangements is 12!/[2! 5!]

Circular Permutations

In general, the number of ways of arranging n objects around a round table is (n-1)!

An easier way of thinking is that we "fix" the position of a particular person at the table. Then the remaining n -1 persons can be seated in (n-1)! ways. Done!

Thus the number of ways of arranging n persons along a round table so that no person has the same two neighbours is(n-1)!/2

Similarly in forming a necklace or a garland there is no distinction between a clockwise and anti clockwise direction because we can simply turn it over so that clockwise becomes anti clockwise and vice versa. Hence the number of necklaces formed with n beads of different colours = (n-1)!/2

Example

1. A cat invites 3 rats and 4 cockroaches for dinner. How many seating arrangements are possible along a round table? 

1. SOLUTION: "Fix" the position of the cat. Now remaining 3 rats and 4 cockroaches can be seated in 7!/(3! 4!) = 35 ways.

Combinations - Formula for ⁿC_r

The number of combinations of n different things taken r at a time is
                   ⁿC_r = n!/[ r! (n -r)!]

Corollaries:

1. ⁿC_r = [n (n -1)(n -2).....upto r factors]/r!
2. ⁿC_n = 1
3. ⁿC₀ = 1
4. ⁿC_r = ⁿC_{n -r}

Combinations - Some important results

1. If out of n different things, any number of items can be chosen, it can be done inⁿC₀ +ⁿC₁ +ⁿC₂ +... +ⁿC_n ways. Alternatively, any of n items may or may not be chosen. Hence number of selections = 2×2×...n times = 2ⁿ
   => ⁿC₀ +ⁿC₁ +... +ⁿC_n = 2ⁿ.
   It can be shown that ⁿC₀ +ⁿC₂ +ⁿC₄ +... = ⁿC₁ +ⁿC₃ +ⁿC₄ +... = 2^n-1.
2. Out of n different things, at least one (or more) can be chosen in 2ⁿ -1 ways.
3. Number of combinations of n different things taken r at a time when p particular things always occur is ^{n -p}C_{r -p}. Number of permutations of these is ^{n -p}C_{r -p}.r!
4. Number of combinations of n different things taken r at a time when p particular things never occur is ^n-pC_r. Number of permutations of these is ^{n -p}C_r.r!
5. The number of ways in which m +n different things can be divided into two groups containing m and n things respectively is (m +n)!/[m!n!]. The reason is that whenever you choose a group of m out of m +n, a group of n is automatically left behind. Number of combinations of m +n things taken m at a time is
             ^{n +m}C_m = (m +n)!/[m! (m +n -1)!] = (m + n)!/[m! n!]
6. If subgroups are equal i.e. n = m, then 2m things can be divided into two groups of m each in (2m)!/(m!)² ways. If you distribute things to two persons, then this formula gives number of subdivisions. If it is possible to interchange the two groups then number of divisions is (2m)!/[(m!)².2!]
7. Number of division of m +n +p things into groups of m, n, p things respectively is (m +n +p)!/[m!.n!.p!]
8. If 3m things are divided into 3 equal groups, then number of divisions is (3m)!/(m!)³ and if the groups are interchangeable, the number of divisions is (3m)!/[(m!)³.3!]
9. If there are p +q +r things, where p things are alike, q things are alike and r things are alike, a non-empty selection can be made in (p +1)(q +1)(r +1) -1 ways as 0, 1, 2, ..., p items of p may be chosen; 0, 1, 2, ..., q items of q may be chosen etc.
10. If there are p +q +r things, where p things are alike, q things are alike and remaining r are all different, then a non-empty selection can be made in (p +1)(q +1). 2r -1 ways. (Prove it!)

Example

In how many ways can final eleven be selected from 15 cricket players if

1. there is no restriction
2. one of them must be included
3. one of them, who is in bad form, must always be excluded
4. two of them being leg spinners, one and only one leg spinner must be included?

Solution

1. 11 players can be selected out of 15 in ¹⁵C₁₁ ways
   = ¹⁵C₄ = (15.14.13.12) / (1.2.3.4) = 1365 ways.
2. Since a particular player must be included, we have to select 10 more out of remaining 14 players. This can be done in
   ¹⁴C₁₀ = ¹⁴C₄ = (14.13.12.11)/(1.2.3.4) = 1001 ways.
3. Since a particular player must be always excluded, we have to choose 11 players out of remaining 14. This can be done in
   ¹⁴C₁₁ ways = ¹⁴C₃ = (14.13.12)/(1.2.3) = 364 ways.
4. One leg spinner can be chosen out of 2 in² C₁ = 2 ways. Then we have to select 10 more players out of 13 (because second leg spinner can't be included). This can be done in
   ¹³C₁₀ ways = ¹³C₃ = (13.12.11)/(1.2.3) = 286 ways.
   Thus required number of combinations = 2×286 = 572

Reasoning Quiz For SBI Pre

Directions (Q. 1-6) : In each of the questions below are given three statements followed by two conclusions numbered I and II . You have to take the given statements to be true even if they seem to be at variance with commonly known facts. Read all conclusions and then decide which of the given conclusions logically follows from the given statements, disregarding commonly known facts.Give answer 
1) if only conclusion I follows
2) if only conclusion II follows
3) if either conclusion I or II follows
4) if neither conclusion I nor II follows
5) if both conclusions I and II follows

(Q. 1 - 2) Statements:
Some poor are rich.
All rich are doctors.
Some intelligents are doctors.

1. Conclusions: I. At least some poor are intelligents.
                         II. All intelligents being rich is a possibility.

2. Conclusions: I. All intelligents being doctors is a possibility.
                         II. Some poor are doctors.

(Q.3 - 4 ) Statements:
All fans are bulbs.
All wires are holders.
Some wires are bulbs.

3. Conclusions: I. At least some fans are wires.
                         II. All holders being fans is a possibility.

4. Conclusions: I. All fans being holdrs is a possibility.
                         II. Some holders are bulbs.

(Q.5 - 6) Statements:
No savings accounts is a current accounts.
Some fixed deposits are savings accounts.
Some current accounts are recurring deposits.

5. Conclusions: I. All savings accounts being current accounts is a posibility.
                           II. All fixed deposits being current accounts is a possibility.

6. Conclusions: I. All current accounts being fixed deposits is a possibility.
                          II. All savings accounts being recurring deposits is a possibility.

Directions (Q. 7-8): Read the following information carefully and answer the questions that follow.
Among five friends - Raj, Vijay, Shadab, Alok and Vishal each friend is heavy weight. Only one friend has more weight than Shadab. Vishal has more weight than Raj but less than Alok. Alok is not the heaviest. The one who is the second lowest has 71 kg weight. Shadab has 92 kg weight.

7. Which of the following is the most likely weight of Vijay?
1) 81 kg           2) 45 kg           3) 58 kg         4) 90kg                    5) None of these
8. Who among the following is most likely to have 87 kg weight?
1) Alok        2) Shadab       3) Vishal    4) can't be determined     5) None of these


Answers :)

1. 2
2. 5
3. 2
4. 5
5. 4
6. 5
7. 5  Vijay > Shadab > Alok > Vish > Raj
8. 1

English Quiz :)

Directions (Q. 1-5): In each of the following questions, a short passage is given with one of the lines in the passage missing and represented by a blank. Select the best out of the five answer choices given to make the passage complete and coherent.

1. It is a sad irony that when international oil prices are on a downward spiral, India has administered its biggest increase ever in the domestic price of petrol. __________________The winds of reform that blew across other infrastructure industries such as telecom, trasforming them unrecognisably, have largely been absent in the oil industry. The result is a mess characterised by ballooning subsidies, opaque pricing policies and a government that is guided more by political expediency than economic considerations in managing the industry.

1) We need greater transparency in the pricing methodology adopted by the oil companies.
2) Behind this irony lies a story of gross mismanagement of the oil economy by successive governments.
3) Behind this we find great cooperation of oil companies.
4) It did not allow oil companies to increase prices in the last six months.
5) Why should domestic prices of the two fuels be linked to their international prices?

2. The quest for the optimal mobile device that can advance education by improving access to learning materials and reducing costs is engaging many countries, including India. Among the more promising gadgets for the task are e-readers, which use e-ink display technology that closely matches the look and feel of a printed black and white page. Tablet computers, on the other hand, offer a full-colour, backlit screen alternative suitable for multimedia and rich content. _____________ and this concept has been put to the test in a field setting in Ghana involving distribution of e-reader hundreds of school students.

1) These technologies can significantly aid the teaching - learning process
2) These ranged from wirelessly delivered sponsored text to open access volumes that they downloaded
3) These results from the project make it clear
4) India’s grand plan to identify a tablet device for educational purposes has been virtually stillborn.
5) The centre has conceded that the Aakash tablet computer has been distributed in limited numbers.

3. Yet it can still be called ‘Premier’, for it speaks for the more prosperous parts of India, ________ the very names of the teams are a clue to its elitist character-two ‘kings’, two ‘Royals’ and one ‘knight’, this in a democratic republic whose constitution and laws did away with aristocratic titles of any kind.

1) and which is shining as well as wining
2) that most closely follow these teams
3) It contains minor section
4) and for the more prosperous sections within them.
5) they don’t go for it.

4. All of us should pause and took at some unpleasant realities that are going to unfold in the future. __________ . We must find ways and means to reduce our dependency on foreign oil. This can only be done by reducing oil consumption, especially for personal transport. The government must devise suitable transport policies that take off the most number of personal vehicles from public roads. If we fail to make suitable adjustments to our lifestyles, the transition to a less carbon intensive era is bound to be chaotic and disruptive. This is time of inflation.

1) The era of cheap and abundant oil is over.
2) There is too much consumption of oil.
3) There is no more consumption of oil.
4) The govt must find ways to reduce the price of fuel.
5) Oil marketing companies are of little relevance.

5. After a gap of over eight decades, the painstaking attempt to enumerate and measure castes has once again got under way. _________ Most of those argument revolved around the means by which the exercise could be accomplished and the possible end results of a cost-based head-count.

1) The current compilation of data takes the form of a socio economic records.
2) Various ambitious castes quickly perceived the chances of raising their status.
3) It has both aspects.
4) Why each caste shouldn’t response.
5) The pros and cons of compiling data on caste have been exhaustively discussed.

Directions (Q.6-10): Rearrange the following seven sentences or part of sentences (A), (B), (C), (D), (F), (F) and (G) in the proper sequence to form a meaningful paragraph; then answer the question given below it.

(A) Since 1988 when the world embarked on an effort to wipe out the disease entirely,
(B) Developed by an American scientist, Albert Sabin, the oral polio vaccine (OPV) uses weakened strains of the virus.
(C) The viruses in the vaccine replicate in cell in the gut and evoke an immune response that protects the child when a wild virus comes along.
(D) The vaccine is easy to administer - simply put a couple of drops of it in a child’s mouth.
(E) Widespread use of the oral vaccine has brought the eradication of polio tantalisingly within reach.
(F) the number of cases has fallen by 99.8 percent.
(G) Wouldn’t it be nice to have a better sort of oral polio vaccine?

6. Which of the following should be the FIRST sentence after rearrangement?
1) A        2) D           3) E          4) F            5) G

7. Which of the following should be the FOURTH sentence after rearrangement?
1) B        2) C           3) D          4) E            5) F

8. Which of the following should be the SECOND sentence after rearrangement?
1) G       2) E            3) F           4) D           5) B

9. Which of the following should be the FIFTH sentence after rearrangement?
1) A      2) B             3) C          4) D            5) B

10. Which of the following should be the SEVENTH (Last) sentence after rearrangement?
1) E       2) D            3)B           4) C             5) A