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

$^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}=\frac{n !}{(n-r) ! r !}$

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

$50 \mathrm{C}_{5}=\frac{50 !}{(50-5) ! 5 !}$

$\begin{array}{l}{=\frac{50 !}{45 ! 5 !}} \\\\ {=\frac{50 \times 49 \times 48 \times 47 \times 46 \times 45 !}{45 ! \times 5 !}}\end{array}$

$\begin{array}{l}{=\frac{50 \times 49 \times 48 \times 47 \times 46}{5 \times 4 \times 3 \times 2 \times 1}} \\\\ {=\frac{50 \times 49 \times 48 \times 47 \times 46}{10 \times 12}}\end{array}$

$\begin{array}{l}{=5 \times 49 \times 4 \times 47 \times 46} \\\\ {=2118760}\end{array}$

Hence, there are 2118760 ways to select.