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# Permutation and Combination Problems with Solutions for IBPS Exam

Permutation and Combination Problems with Solutions with solution for SSC Exam

Permutation Permutation is the total number of different ways of arrangements

Number of ways of arrangements of n different things = n!

For example In how many different ways can the letters of word ‘TIGER’ be arranged

Solution Since there are five different letters, so total arrangements = 5! = 120

But if the letters or numbers are repeating suppose there are n numbers out of which two numbers are repeated 2 times and 3 times respectively. Then total arrangements = n! / (2!×3!)

For example Total arrangements of the letters of the word ‘committee’ is equal to

9! /( 2!×2!× 2!) = 45360

Number of ways of arrangements of r things taken at a time out of n different things = nPr

nPr = n! / (n-r)!

For example How many three digits number can be formed with the digits 1,2,3,4,5,6, if repetition of digits is not allowed

Solution 6P3 = 6 ! / (6 – 3) ! = 6! / 3! = 720 / 6 = 120

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Combination Combination is the total number of ways of selecting  or choosing total number of ways of selecting r things taken at a time  out of n different things = nCr

nCr = n! / [r! (n – r)!]

For example In how many different ways can we choose 4 persons out of 6 persons

Solution 6C4 = 6! / 4! (6 – 4)!   = 6! / (4! 2!) = 6*5*4*3*2*1 / 4*3*2*1*2*1 = 6*5 / 2 = 15

Total number of different ways of arrangements of n persons around a circular table = ( n – 1 )!

In permutation and combination ‘or’ means +,  ‘and’ means

Suppose we have to form a group of 3 men and 1 women out of  5 men and 4 women

Total number of ways of forming this group = 5C3 and 4C1

= 5C3 × 4C1 = 10*4 = 40

Now suppose we have to form a group of 2 people of same gender ( i.e. both men or  both women) out of 5 men and 4 women

Total number of ways = 5C2 or 4C2

=   5C3 + 4C2 = 10 + 6 = 16

Probability

Experiment : An activity which results in some well defined outcomes is called an experiment.

Random Experiment : An experiment whose outcomes cannot be predicted is called random experiment

Here are some examples of random experiment

Tossing of a coin If we toss a coin either a head or tail will come up. But we cannot predict with certainty out of head or tail before our results

Drawing a card from a well shuffled pack of 52 playing cards

Throwing an unbiased dice If we throw a dice then the possible outcomes are 1, 2, 3, 4, 5, 6

There are total of 52 cards

Types                        number of cards                             color

Heart                           13 cards                                         red

Club                             13 cards                                         black

Diamond                     13 cards                                         red

2,3,4,5,6,7,8,9,10, Jack , Queen, King and Ace are the thirteen cards of each type

Probability The probability of any event E is calculated as favorable event divided by the total event

P (E)  = favorable event / total event

Practice set problems

1. Find the total arrangements of the letters of word ‘ MISSISSIPPI ‘

a. 34650         b. 32540          c. 28450          d. 24560        e. None of these

2. Out of 5 men and 3 women, a committee of 3 members is to be formed so that it has 1 woman and 2 men. In how many different ways can it be done ?

a.20         b. 10         c.23         d. 30           e. None of these

United Bank Of India PO Exam. 21.06.2009

3. In how many different ways can the letters of the word DESIGN be arranged so that the             vowels are at the two ends ?

a. 48        b. 72        c. 36         d.24          e. None of these

United Bank Of India PO Exam. 21.06.2009

4. In how many different ways can the letters  of the word PUNCTUAL  be arranged ?

a.64         b. 40320         c. 960         d. 20160         e. None of these

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SBI PO Preliminary exam. 27.07.200

5. In how many different ways can the letters of the word  CORPORATION be arranged in such a way that the vowels always come together ?

a.840        b. 86400        c. 8400         d. 1440          e. None of these

SBI BANKS PO EXAM. 18.05.2003

6. A committee of five members is to be formed out of 4 students, 3 teachers and 2 sports coaches. In how many can the committee be formed such that the committee should consist of any five people ?

a. 126         b.45          c. 120          d. 24            e. None of these

United Bank Of India PO Exam. 14.11.2010

7. Find the total arrangements of the letters of the word ‘INVISIBILITY’ such that all I always come together

a. 7! × 5!         b. 8!         c.8! × 5!         d. 7!           e. none of these

8. A group of 7 persons is to be formed from 6 men and 8 women such that the men are in majority. In how many ways this can be done ?

a. 780         b. 1260        c. 1016          d. Cannot be determined       e. None of these

9. How many 7_ digits numbers can be formed using the digits 1,2,3,4,5,0,6 such that unit place is 6 and repetition of digits is not allowed ?

a.720         b. 600          c. 120          d. 7!          e. None of these

10. How many even numbers can be formed using the digits 1,2,4,6,8 without repetition of digits ?

a. 120        b. 96         c. 56          d. Cannot be determined  e. None of these

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11. How many 4 digits odd numbers can be formed using the digits 0,1,2,3 if repetition is allowed ?

a. 96         b. 256         c. 192         d. 81         e. None of these

12. Find the total arrangements of the letters of the word APPLE such that A and P does not come together

a. 18               b. 60           c. 45              d. 30               e. None of these

13. A basket contains 6 red, 5 green and 8 blue balls. If four balls are picked at random, what is the probability that all four of them are either red or any two of them are either red or any two out of the four are green ?

a. 5 / 1292         b. 925 / 3876         c. 359 / 1938             d. 11 / 3876

e. none of these

Union bank of India PO 27.11.2005

14. In a container there are 28 eggs out of which 8 eggs are rotten. If two eggs are chosen at random, what will be the probability that at least one egg is rotten ?

a. 94/189            b. 95/187               c. 93/189             d. 97/189               e. None of these       LIC Assistant Administrative Officer 12.05.2013

15. A bag contains 2 red, 3 green and 2 blue balls. 2 balls are to be drawn randomly.What is the probability that the balls drawn contain no blue ball ?

a. 5/7           b. 10/21              c. 2/7        d. 11/21         e.None of these

16. If the probability of any event ‘A’ is 0.995, then the probability of Ac ( not having ‘A’ ) is

a. 0.05           b. 0.005          c. 0.105           d. 0.5              e. 0.1

17. From a well shuffled deck of cards, two cards are choosing randomly. Find the probability that both the cards are king or both are queen.

a. 1/121            b. 1/221         c. 2/221      d. 3/221       e. None of these

18. If a dice is thrown thrice, what is the probability that the sum of the number appearing on the dice is greater than 15 ?

a. 5/108        b. 7/216        c. 4/216        d. 5/216        e. None of these