## Permutation and Combination Problems with Solutions

# 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 = nP^{r}

nP^{r} = 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** 6P^{3} = 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 = nC^{r}

nC^{r} = n! / [r! (n – r)!]

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

**Solution **6C^{4} = 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 = 5C^{3} and 4C^{1}

= 5C^{3} × 4C^{1} = 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 = 5C^{2} or 4C^{2}

= 5C^{3} + 4C^{2} = 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

**Information about playing cards **

There are total of 52 cards

Types number of cards color

Spade 13 cards black

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** **

**Following Question and Solution Prepared by Best Bank Coaching in Noida**

** 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

**U****nion 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

**RBI Grade B Officer 17.11.2002**

16. If the probability of any event ‘A’ is 0.995, then the probability of A^{c} ( 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

** Answer key**

1.A 2.D 3.A 4.D 5.E 6.A 7.B 8.C 9.B 10.B 11.A 12.D 13.B 14.A 15.B 16.B 17.C 18.A

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