How To Solve Combination Questions Quickly

Type 1: How to Solve Combination Question (with or without repetition)

Question 1.There was a flock of sheep in which 6 were male sheep 5 were female sheep. Now we need to select 4 sheep to take out wool from them. In how many different ways can they be selected such that at least one male sheep should be there?

Options:

A. 1450

B. 302

C. 295

D. 154

Solution:     The selection can be made in following manner

(1 male sheep and 3 female sheep) or (2 male sheep and 2 female sheep) or (3 male sheep and 1 female sheep) or (4 male sheep)
Required number of ways = (₆C₁ * ₅C₃) + (₆C₂ * ₅C₂) + (₆C₃ * ₅C₁) + (₆C₄)

Required number of ways = (6 * 5) + (15 * 10) + (20 * 5) + 15

Required number of ways = 30 + 150 + 100 + 15

Required number of ways = 295

Correct option: C

Question 2.Among a set of 5 white balls and 3 blue balls, how many selections of 5 balls can be made such that at least 3 of them are white balls.

Options:

A. 45

B. 46

C. 44

D. 40

Solution:    The selection can be made in following manner

(3 white ball and 2 blue ball) or (4 white ball and 1 blue ball) or (5 white ball)

₅C₃ * ₃C₂ + ₅C₄ * ₃C₁ + ₅C₅

(10 * 3) + (5 * 3) + 1

 30 + 15 + 1

46

Correct option: B

How To Solve Quickly Combination Questions

Question 3. There are 7 consonants and 4 vowels. Find out how many words of 3 consonants and 2 vowels can be formed?

Options:

A. 120

B. 102

C. 20

D. 210

Solution:    Number of ways of selecting 3 consonants from 7= ₇C₃

Number of ways of selecting 2 vowels from 4= ₄C₂

Number of ways of selecting 3 consonants from 7 and 2 vowels from 4 = ₇C₃ × ₄C₂ = 35 * 6 = 210

Correct option: D

Question 4. In how many ways can a team of 5 cricketers can be formed out of a total of 10 cricketers such that two particular cricketers should be included in each team?

Options:

A. 66

B. 65

C. 56

D. 22

Solution:     Two particular cricketers should be included in each team.

Therefore, select remaining 5-2=3 cricketers from 10-2=8 cricketers = ₈C₃

₈C₃ = \frac{8!}{(8-3)! 3! } = 56

Correct option: C

Question 5. There are 3 types of t-shirts available in a clothing store. In how many ways can 5 t-shirts be selected?

Options:

A. 21

B. 42

C. 5

D. 12

Solution:     ^{r + n – 1}C_r = ^{5 + 3 – 1}C₁₅ = ⁷C₅

We know that, 

^nC_r = \frac{n!}{(n-r)! r! }

⁷C₅ = \frac{7!}{(7-5)! 5! } = 21

Correct option: A