Answer
$4,516,932,420$ committees
Work Step by Step
Committee members are chosen with no importance of order of choice, so we deal with combinations.
We have a sequence of selections in which we choose
1. ... 4 out of a group of 55 Republicans ... in ${}_{55}C_{4}$ ways
2. ... 3 out of a group of 44 Democrats... in ${}_{44}C_{3}$ ways
By the Fundamental Counting Principle,
Total ways= ${}_{55}C_{4}\cdot {}_{44}C_{3}$
${}_{55}C_{4}=\displaystyle \frac{55!}{(55-4)!4!}$
$=\displaystyle \frac{55\times 54\times 53\times 52}{1\times 2\times 3\times 4}=341,055$
${}_{44}C_{3}=\displaystyle \frac{44!}{(44-3)!3!}=\frac{44\times 43\times 42}{1\times 2\times 3}=13,244$
Total=$341,055\times 13,244$= $4,516,932,420$
