Respuesta :
Answer:
There 54,600 ways to form a committee with six members with the same number of men and women.
Step-by-step explanation:
Since we are to form a committee of six members with the same number of men and women, that is
In the committee, we must have 3 men and 3 women.
First we will find the number of ways to form a committee of 3 men out of 10 men and then 3 women out of 15 women, we will then use the product rule to determine the number of ways to form the committee of 3 men and 3 women.
From the combination formula
[tex]^{n} C_{r} = \frac{n!}{(n-r)!r!}[/tex]
Where n is the number of items is set
and r is the the number of items selected from set
To form a committee of 3 men out of 10 men
Then,
n = 10
and r = 3
∴ [tex]^{10} C_{3} = \frac{10!}{(10-3)!3!}[/tex]
[tex]^{10} C_{3} = \frac{10\times9\times8\times 7!}{7!3!}[/tex]
[tex]^{10} C_{3} = \frac{10\times9\times8}{3!}[/tex]
[tex]^{10} C_{3} = \frac{10\times9\times8}{6}[/tex]
[tex]^{10} C_{3} = 120[/tex]
Then,
To form a committee of 3 women out of 15 men
n = 15
r = 3
∴ [tex]^{15} C_{3} = \frac{15!}{(15-3)!3!}[/tex]
[tex]^{15} C_{3} = \frac{15\times14\times13\times 12!}{12!3!}[/tex]
[tex]^{15} C_{3} = \frac{15\times14\times13}{3!}[/tex]
[tex]^{15} C_{3} = \frac{15\times14\times13}{6}[/tex]
[tex]^{15} C_{3} = 455[/tex]
Applying the product rule: If there are n ways to do A and m ways to do B, then the number of ways to do A and B is n × m.
Therefore, the number of ways to form a six members committee of 3 men and 3 women is
120 × 455 = 54600
Hence, there 54,600 ways to form a committee with six members with the same number of men and women.
Using the combination formula, it is found that there are 54,600 ways to form the committee.
The order in which the members of the committee are chosen is not important, hence the combination formula is used to solve this question.
Combination formula:
[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by:
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
In this problem, the committee is formed by:
- 3 men from a set of 10.
- 3 women from a set of 15.
Hence, the number of ways is:
[tex]T = C_{10,3}C_{15,3} = \frac{10!}{3!7!} \times \frac{15!}{3!12!} = 120 \times 455 = 54600[/tex]
There are 54,600 ways to form the committee.
A similar problem is given at https://brainly.com/question/24437717
