Question

A city council consists of six democrats and five republicans. If a committee of seven people is selected, find the probability of selecting four Democrats and three republicans.

Answer 1

Use hypergeometric distribution where there are two categories of identical objects/persons, each with a know size.
d=number of Democrats selected
D=total number of Democrats = 6
r=number of Republicans
R=total number of Republicans =5
Then 
$P(d,r)=\frac{C(D,d)C(R,r)}{C(D+R,d+r)}$
where 
$C(n,r)=\frac{n!}{(n!(n-r)!)}$  = combination of r items selected from n,
D+R=total number of members = 6+5 =11
d+r=number of members selected = 7

$P(d,r)=\frac{C(D,d)C(R,r)}{C(D+R,d+r)}$
$P(4,3)=\frac{C(6,4)C(5,3)}{C(6+5,4+3)}$
$=\frac{C(6,4)C(5,3)}{C(11,7)}$
$=\frac{15*10}{330}$
$=\frac{5}{11}$

Answer: the probability of selecting 4 Democrats and 3 Republicans is 5/11