Question

A certain club has 20 members. What is the ratio of the number of 5-member committees that can be formed from the members of the club to the number of 4-member committees that can be formed from the members of the club?

Answer 1

Answer:

The ratio is 16:5

Step-by-step explanation:

In order to find this ratio, you need to first find out in how many ways you can create 5-member committees and 4-member committees. But before doing so, you need to determine if the order in which you pick the members for the committee, matters or not.

In this case, the order doesn't matter, which means we can use a combination to calculate them. Combinations are found by using the following formula:

$_{n}C_{r}=\frac{n!}{r!(n-r)!}$

Where n is the number of available items and r is the number of desired items in the group.

So, for the 5-member committees, n will be 20 and r is 5, so the formula will look like this:

$_{20}C_{5}=\frac{20!}{5!(20-5)!}$

which can now be simplified, so we get:

$_{20}C_{5}=\frac{20!}{5!(15)!}$

So now we can solve this:

$_{20}C_{5}=\frac{20*19*18*17*16*15!}{5*4*3*2*1(15)!}$

The 15!'s get cancelled so I get:

$_{20}C_{5}=\frac{20*19*18*17*16}{5*4*3*2*1}=15,504$ possible groups with 5 members each.

So we can find the number of 4-member committees in the same way, with the difference that r=4 this time. So we get:

$_{20}C_{4}=\frac{20!}{4!(20-4)!}$

$_{20}C_{4}=\frac{20!}{4!(16)!}$

which solves to:

$_{20}C_{5}=4845$

So now that we have the number of possible 5-member committees and the number of possible 4-member committees, we can find the ratio between them, which we will get by dividing the numbers:

15504:4845

which simplifies to

16:5

which means that tere are 16 5-member committees for every 5 4-member commitees.