Question

a legislative committee consists of 7 democrats and 4 republicans. a delegation of 3 is to be selected to visit a small island republic. complete parts (a) through (d) below
(a) how many different delegations are possible?
the 3 delegates can be selected 165 different ways.
(b) how many delegations would have all democrats?
the 3 delegates can be selected 35 different ways if all 3 are democrats.
(c) how many delegations would have 2 democrats and 1 republican?
the 3 delegates can be selected □ different ways if 2 are democrats and 1 is a republican.

Explanation:

Step1: Calculate 2 Democrats from 7

We use combinations: $\binom{n}{k}=\frac{n!}{k!(n-k)!}$. For choosing 2 Democrats from 7:
$\binom{7}{2}=\frac{7!}{2!(7-2)!}=\frac{7\times6}{2\times1}=21$

Step2: Calculate 1 Republican from 4

For choosing 1 Republican from 4:
$\binom{4}{1}=\frac{4!}{1!(4-1)!}=4$

Step3: Multiply the two results

Multiply the number of ways to choose Democrats and Republicans:
$21\times4=84$

Answer:

84