Respuesta :
Answer:
Number of committees that are possible =210
Step-by-step explanation:
Total names in ballot = 10
Total members to be elected for the committee = 4
So, we need the number of different ways in which 4 names out of 10 can be elected for the committee.
To find the number of ways in which the committee can be elected, we use the combinations which is given as
[tex]aCb=\frac{a!}{(a-b)!b!}[/tex]
where [tex]a[/tex] represents total number of members and [tex]b[/tex] represents total number of members to be elected.
So, [tex]10C4=\frac{10!}{(10-4)!4!}[/tex]
[tex]10C4=\frac{10!}{6!4!}[/tex]
[tex]10C4=\frac{10\times9\times8\times7\times6\times5\times4\times3\times2\times1}{6\times5\times4\times3\times2\times1\times4\times3\times2\times1}[/tex]
[tex]10C4=\frac{10\times9\times8\times7}{4\times3\times2\times1}[/tex] [Canceling the common numbers]
[tex]10C4=210[/tex]
∴ Number of committees that are possible =210
Using the combinations formula, we will see that 210 different committees are possible.
How many different committees are possible?
Here we need to use the combinations function. It says that for a set of N elements, the number of different subsets of K elements (such that K is equal or smaller than N) that we can make is:
[tex]C(N, K) = \frac{N!}{(N - K)!*K!}[/tex]
In this case, we have N = 10 and K = 4, replacing that we will get:
[tex]N(10, 4) = \frac{10!}{(10 - 4)!*4!} = \frac{10!}{6!*4!} = \frac{10*9*8*7}{4*3*2} = 210[/tex]
This means that from 10 names, there are 210 different committees of 4 members that can be made.
If you want to learn more about combinations you can read:
https://brainly.com/question/25821700
