# Permutation and Combination - Aptitude Questions and Answers

## Permutation and Combination Questions and Answers

Learn and practice the chapter "Permutation and Combination" with these solved Aptitude Questions and Answers. Each question in the topic is accompanied by a clear and easy explanation, diagrams, formulae, shortcuts and tricks that help in understanding the concept.

## Use of Permutation and Combination Questions

The questions and examples given in this section will be useful to all the freshers, college students and engineering students preparing for placement tests or any competitive exam like MBA, CAT, MAT, SNAP, MHCET, XAT, NMAT, GATE, Bank exams - IBPS, SBI, RBI, RRB, SSB, SSC, UPSC etc.

Practice with this online test to crack your placements and entrance tests!
1. Arrange the letters of the word "DARKER" so that the three vowels do not appear together. In how many ways can this be done?

a. 240
b. 360
c. 500
d. 720

Explanation:

'DARKER' has 6 letters.

Tip:
We can arrange 'n' things in n! ways.

While arranging letters/things/numbers, if there are two same things or things get repeated twice, then we need to divide by 2!
If there are 3 same things or things get repeated thrice, then divide by 3! And so on..

Thus, we can arrange 6 letters in 6! ways.
But R gets repeated. There are 2 Rs. So divide by 2!
 ∴ Total ways = 6! = 360 2!
Vowels not together = Total ways - Vowels together

Consider the 2 vowels (A and E) as one group.
We have 4 letters and 1 group = 5
We can arrange them in 5! Ways.
 But again here R comes twice. So we will have 5! 2!
Also, the 2 vowels can be arranged in 2! Ways.
 So, Number of ways with vowels together = 2! x 5! = 120 2!
∴ Number of ways with vowels not together = 360 - 120 = 240

2. In a box there are coins marked 0, 2, 3, 4, 5, 6, 8. Without repeating any digit, how many numbers can you form in the range 500 - 1000 (excluding 500 and 1000).

a. 60
b. 70
c. 90
d. 147

Explanation:

There are total 7 digits given - 2, 3, 4, 5, 6, 8 and 0
Between 500 and 1000 means from 501 to 999.
All of them are 3 digit numbers.

First digitSecond digitThird digit
5, 6 or 8 i.e. 3 possibilities (1 digit gets used here)Any one from 7-1 = 6 remaining digits i.e. 6 possibilitiesAny one from 6-1 = 5 remaining digits i.e. 5 possibilities

∴ Total numbers possible = 3 x 6 x 5 = 90

3. There are 8 friends sitting in a cafĂ©. In how many ways can you form a group of 4 people so that

i. Two particular friends are definitely there
ii. Two particular members are definitely not there

a. 15 and 15
b. 15 and 360
c. 30 and 360
d. 360 and 360

Explanation:

Tip:
 SELECT = Combination = nCr = n! r!(n-r)!
 SELECT and ARRANGE = Permutation = nPr = n! (n-r)!

I. Compulsorily include 2 particular members.
When we include these 2 members, we are left with 4 - 2 = 2 spots in the group
Also, number of members which remain are 8 - 2 = 6
So now we need to select 2 people out of 6.
 6C2 = 6! = 15 = number of ways 2!4!

II. Compulsorily exclude 2 particular members.
When we exclude these 2 members, we are left with 8 - 2 = 6 members
So now we need to select 4 people out of 6.
 6C4 = 6! = 15 = number of ways 4!2!

4. Priyanka want to attend Grammy awards at London and National awards at Delhi before heading to the UNICEF Summit in Berlin. 8 flights come from London to Delhi through different routes and 6 flight fly from Delhi to Berlin through 6 different routes. Find the number of ways in which Priyanka can reach from London to Berlin through Delhi.

a. 14
b. 24
c. 48
d. 100

Explanation:

This is very easy to solve
Say out of 8, Priyanka chooses one way to go from London to Delhi.
Now, From Delhi to Berlin she has 6 routes i.e. 6 options
Similarly, for 2nd route between London and Delhi, Priyanka will again have 6 options from Delhi to Berlin.
This is true for all 8 routes between London and Delhi.
So, answer = 8 x 6 = 48 possible ways to travel from London to Berlin via Delhi

5. Out of the 35 players attending the opening ceremony of an event there are 12 cricketers, 10 wrestlers, 5 badminton players and 8 hockey players. The management wants a cricketer or a wrestler to lead the group in the parade. In how many ways can this be done?

a. 1/2
b. 13
c. 22
d. 35