The Senate in a certain state is comprised of 55 ​Republicans, 44 ​Democrats, and 1 independent. How many committees can be formed if each committee must have 2 Republicans and 2 ​Democrats?

Answer 1

combinations of 55 taken 2 at a time

time combinations of 44 taken 1 at a time

=(55C2)(44C1)

= (55!/2!53!)(44!/1!43!)

= (55x54x53!/2x1x53!)(44x43!/1x43!)

= (55x54/2)(44)

= 55x27x44

= 65,340 ways

to select 2 republicans and 1 democrat from a Senate with 55 republicans, 44 democrats and 1 independent

use combinations, not permutations if order doesn't matter

for a committee of 2 republicans and 2 democrats

(55C2)(44C2)

= (55!/2!/53!)(44!/2!42!)

=(55x27)(44x43/2)

=55x27x22x43

= 1,404,810 ways