Question

There are 100 members of the U.S. Senate with 2 members from each state. In how many ways can a committee of 5 senators be formed if no state may be represented more than once?

Answer 1

There are 2118760 ways to select a committee of 5 senators be formed if no state may be represented more than once

Solution:

Given that, There are 100 members of the U.S. Senate with 2 members from each state.

Which means there are senates from 100/2 = 50 states.

We have to find in how many ways can a committee of 5 senators be formed if no state may be represented more than once?

As no state can be represented more than once, we just have to take 1 from each state for selections.

So, now we will have 50 senators out of who we have to pick 5 senators.

As we just have to select the senators. We can use combinations here.

In combinations, to pick r items from n items, there will be
`^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}` ways

`^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}=(n !)/((n-r) ! r !)`

Then, here we have to pick 5 out of 50:

`50 \mathrm{C}_(5)=(50 !)/((50-5) ! 5 !)`

`\begin{array}{l}{=(50 !)/(45 ! 5 !)} \\\\ {=(50 * 49 * 48 * 47 * 46 * 45 !)/(45 ! * 5 !)}\end{array}`

`\begin{array}{l}{=(50 * 49 * 48 * 47 * 46)/(5 * 4 * 3 * 2 * 1)} \\\\ {=(50 * 49 * 48 * 47 * 46)/(10 * 12)}\end{array}`

`\begin{array}{l}{=5 * 49 * 4 * 47 * 46} \\\\ {=2118760}\end{array}`

Hence, there are 2118760 ways to select.