In how many ways a group of 4 girls and 7 boys can be chosen out of 10 girls and 12 boys?

In mathematics, permutation is known as the process of arranging a set in which all the members of a set are arranged into some series or order. The process of permuting is known as the rearranging of its components if the set is already arranged. Permutations take place, in more or less important ways, in almost every area of mathematics. They frequently appear when different commands on certain finite sets are considered.

Permutation Formula

In permutation r things are selected from a set of n things without any replacement. In this order of selection matter.

ⁿp_r = (n!)/(n – r)!

n = set size, the total number of items in the set

r = subset size, the number of items to be selected from the set

Combination

A combination is an act of choosing items from a group, such that (not like permutation) the order of choice does not matter. In smaller cases, it is possible to count the number of combinations. Combination refers to the union of n things taken k at a time without repetition  In combination you can select the items in any order. To those combinations in which re-occurrence is allowed, the terms k-selection or k-combination with replication are frequently used.

Combination Formula

In combination r things are selected from a set of n things and where the order of selection does not matter

ⁿC_r =n! ⁄ ((n-r)! r!)

Here, n = Number of items in set

r = Number of items selected from the set 

In how many ways a group of 4 girls and 7 boys can be chosen out of 10 girls and 12 boys?

Solution:

Here we use the formula for selecting r things from n different things.

Complete step-by-step answer:

The number of ways of selecting r things from n different things is given by ⁿC_rand

the value of ⁿC_r is given as ⁿC_r = n! ⁄ r!(n−r)! .

Here sequence doesn’t matter

The formula of selecting 4 girls from 10 girls: ¹⁰C₄

= 10! ⁄ 4!(10-4)!

= 10! ⁄ 4!6!

=10 × 9 × 8 × 7 × 6!⁄ 4 × 3 × 2 × 1 × 6!

= 10 × 9 × 8 × 7⁄4 × 3 × 2

= 10 × 9 × 8 × 7⁄8 × 3

= 10 × 9 × 7⁄3

= 10 × 3 × 76

= 210

The formula of selecting 7 boys from 12 boys: ¹²C₇

= 12! ⁄ 7!(12-7)!

= 12! ⁄ 7!5!

= 12 × 11 × 10 × 9 × 8 × 7! ⁄ 7! × 5 ×  4 × 3 × 2 × 1

= 12 × 11 × 10 × 9 × 8 ⁄ 10 × 12

= 11 × 9 × 8

= 792

No of ways to make the team of 4 girls and 7 boys = ¹⁰C₄ × ¹²C₇

= 210 × 792

= 1,66,320 ways

Similar Questions

Question 1: A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected, if the team has at least one boy and one girl.

Solution:

From a group of 4 girls and 7 boys choose 5 members, such that

there is at least one boy and one girl in the team.

So, first let’s select one boy and one girl. The number of ways of selecting 5 number 

is given as ⁷C₁ × ⁴C₁ = 7×4=28.

Now, the abide three members of the team can be either girls or boys . So, we have to

select three members from a group of nine persons i.e., three girls and six boys.

So, number of ways of selection is given as ⁹C₃=9! ⁄ 3!×6!=9×8×7 ⁄ 6=84

So, the number of ways of  choosing a team of 5 members from a group of 4 girls and

7 boys, such that there is at least one boy and one girl in the team is given as

⁷C₁×⁴C₁×⁹C₃ = 28×84 = 2352

Hence, there are 2352 ways of choosing a team of 5 members from a group of 4 girls and

7 boys, such that there is at least one boy and one girl in the team.

Question 2: A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has no girl.

Solution:

We have to choose a team of 5 members from a group of 4

girls and 7 boys, such that there are no girls in the team, i.e. all the members should

be boys.

So, we have to select 5 boys from a group of 7 boys.

So, number of ways =⁷C₅

=7! ⁄ 5!(7−5)!

=7! ⁄ 5!×2!

Now, we know, we can write n! as n(n−1)!=n(n−1)(n−2)! . So, 7!=7×6×5!

So, the number of ways =7×6×5! ⁄ 5!×2!=7×6 ⁄ 2=21.

Therefore , there are 21 ways of choosing  a team of 5 members from a group of 4 girls 

and 7 boys, in such a way that there are no girls in the team.

Question 3: In how many ways of 4 girls and 1 boy, can be chosen out of 4 girls and 7 boys to make the team?

Solution:

Here we use the formula for selecting r things from n different things.

Complete step-by-step answer:

The number of ways of selecting r things from n different things is given by ⁿC _rand

the value of ⁿC_r is given as ⁿC_r=n! ⁄ r!(n−r)! .

Here sequence doesn’t matter

The formula of selecting 4 girls from 4 girls :- ⁴C₄

= 4! ⁄ 4!(4-4)!

= 4! ⁄ 4! 0!

=1

The formula of selecting 1 boys from 7 boys  :- ⁷C₁

= 7! ⁄ 1!(7-1)!

=7! ⁄ 1! 6!

= 7 × 6! / 6!

= 7

No of ways to make the team of 4 girls and 1 boys = ⁴C₄   × ⁷C₁

= 1 × 7

 = 7 ways

Question 4: In how many ways of 3 girls and 2 boys, can be chosen out of 4 girls and 7 boys to make the team?

Solution:

Here we use the formula for selecting r things from n different things.

Complete step-by-step answer:

The number of ways of selecting r things from n different things is given by ⁿC_r and

the value of ⁿC_r is given as ⁿC_r=n! ⁄ r!(n−r)! .

Here sequence doesn’t matter

The formula of selecting 3 girls from 4 girls :- ⁴C₃

= 4! ⁄ 3!(4-3)!

= 4! ⁄ 3! 1!

=4 × 3! / 3!

= 4

The formula of selecting 2 boys from 7 boys  :- ⁷C₂

= 7! ⁄ 2!(7-2)!

=7! ⁄ 2! 5!

= 7 × 6 × 5! / 2! 5!

= 7 × 6 / 2

= 7 × 3

=21

No of ways to make the team of 3 girls and 2 boys = ⁴C₃  × ⁷C₂

= 4 × 21

= 84 ways