Permutation and Combination Solved Questions and Answers (original) (raw)
Last Updated : 30 Jul, 2025
Permutation is defined as the Arrangement of items where order matters, whereas Combination is defined as the selection of items where order doesn’t matter.
**Permutations and Combinations **questions and answers are provided below for you to learn and practice.
**Question 1: How many words can be formed by using 3 letters from the word "DELHI"?
**Solution:
Here we will use the Permutations for this question. Formula for permutation is nPr, for this we have,
n = 5: Total 5 Letters
r = 3: Letters word we required
nPr = n!/(n-r)!
5P3 = 5!/2! = 120/2 = 60
So, Total we can form 60 different permutation of word from Letter Delhi.
**Question 2: How many words can be formed by using the letters from the word "DRIVER" such that all the vowels are always together?
**Solution:
In these types of questions, we assume all the vowels to be a single character, i.e., "IE" is a single character.
So, now we have 5 characters in the word, namely, D, R, V, R, and IE. But, R occurs 2 times.
Number of possible arrangements = 5! / 2! = 60
Now, the two vowels can be arranged in 2! = 2 ways.
Total number of possible words such that the vowels are always together = **60 × 2 = **120
**Question 3: In how many ways, can we select a team of 4 students from a given choice of 15?
**Solution:
Number of possible ways of selection = 15C4 = 15 ! / ((4 !) × (11 !))
Number of possible ways of selection = (15 × 14 × 13 × 12) / (4 × 3 × 2 × 1) = **1365
**Question 4: In how many ways can a group of 5 members be formed by selecting 3 boys out of 6 boys and 2 girls out of 5 girls?
**Solution:
Number of ways 3 boys can be selected out of 6 = 6 C 3 = 6 ! / [(3 !) **× (3 !)] = (6 **× 5 **× 4) / (3 **× 2 **× 1) = 20
Number of ways 2 girls can be selected out of 5 = 5 C 2 = 5 ! / [(2 !) **× (3 !)] = (5 × 4) / (2 × 1) = 10
Therefore, total number of ways of forming the group = 20 x 10 = **200
**Question 5: How many words can be formed by using the letters from the word "DRIVER" such that all the vowels are never together?
**Solution:
We assume all the vowels to be a single character, i.e., "IE" is a single character.
So, now we have 5 characters in the word, namely, D, R, V, R, and IE.
But, R occurs 2 times.
Number of possible arrangements = 5! / 2! = 60
Now, the two vowels can be arranged in 2! = 2 ways.
Total number of possible words such that the vowels are always together = 60 x 2 = 120 ,
Total number of possible words = 6! / 2! = 720 / 2 = 360
Therefore, the total number of possible words such that the vowels are never together **240
**Question 6: How many words can be formed by using 4 letters from the word "COMPUTER"?
**Solution:
For this question, we will use Permutations.
n=8: Total 8 letters in the word "COMPUTER"
r=4: We are required to form a 4-letter word
Using the permutation formula:
r = \frac{n!}{(n - r)!}
8P4 = \frac{8!}{(8 - 4)!} = \frac{8!}{4!} = \frac{40320}{24} = 1680
So, the total number of different permutations of words we can form from the letters in "COMPUTER" is **1680.
**Question 7: How many words can be formed by using the letters from the word "BALLOON" such that all the vowels__(A and both O's)_ are always together?
**Solution:
We will assume all vowels (A and O) together as a single unit, i.e., "AO" becomes a single character.
Now, we have the characters B, L, L, O, N, and AO. But, L occurs 2 times.
Number of possible arrangements = \frac{5!}{2!} = \frac{120}{2} = 60
Now, the vowels (A and O) can be arranged in 3!=3 ways.
Therefore, the total number of possible words where the vowels are always together is:
60 \times 3 = 180
So, the total number of possible words is **180.
**Question 8: In how many ways can we select a team of 5 students from a given choice of 20?
**Solution:
The number of possible ways of selection is given by:
20C5 = \frac{20!}{5!(20 - 5)!} = \frac{20!}{5! \times 15!}
Simplifying:
\frac{20 \times 19 \times 18 \times 17 \times 16}{5 \times 4 \times 3 \times 2 \times 1} = 15504
So, the number of ways to select 5 students from 20 is **15504.