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 □ different ways.

Explanation:

Step1: Calculate total committee size

Total members = 7 + 4 = 11

Step2: Apply combination formula

We use combinations since order doesn't matter for the delegation. The formula for combinations is $\binom{n}{k} = \frac{n!}{k!(n-k)!}$, where $n=11$ and $k=3$.
$$\binom{11}{3} = \frac{11!}{3!(11-3)!} = \frac{11\times10\times9}{3\times2\times1}$$

Step3: Compute the final value

$\frac{11\times10\times9}{3\times2\times1} = 165$

Answer:

165