Choosing committees

[Jon1234]

a

 

[soroban]

Hello, Jon1234!

The ski club at Tula Tech has 35 members (15 females and 20 males).
A committee of 3 members – a President, a Vice President, and a Treasurer – must be chosen.

1. How many different 3-member committees can be chosen?

There are 35 choices for the President, 34 choices for the V.P., ad 33 choices for the Treasurer.

There are: .$\displaystyle 35\cdot34\cdot33 \:=\:39,\!270$ possible committees.

How many different 3-member committees can be chosen
. . if the president must be female?

There are 15 choices for the female president,
. . 34 choices for the V.P.
. . 33 choices for the Treasurer.

There are: .$\displaystyle 5\cdot 34 \cdot33 \:=\:16,\!830$ such committees.

How many different three-member committees can be chosen
. . if the committee cannot have all females or all males?

There are: .$\displaystyle 15\cdot14\cdot13 \:=\:2,\!730$ all-female committees.
There are: .$\displaystyle 20\cdot19\cdot18 \:=\:6,\!840$ all-male committees.

Hence, there are: .$\displaystyle 2,\!730 + 6,\!840 \:=\:9,\!570$ one-gender committees.

Therefore, there are: .$\displaystyle 39,\!270 - 9,\!570 \:=\:29,\!700$ mixed-gender committees.