Question

A political polling organization wishes to select a smaller focus group from a group of 7 Republicans, 10 Democrats, and 2 Independents. In how many ways can the group be chosen if: (a) it will consist of 1 Republican and 4 Democrats?(b) it will consist of 2 Republicans, 2 Democrats, and 3 Independents?

Answer 1

a) 1470 ways

b) 2835 ways

Explanation:

the question is not correct, in the correct question The number of independents should be 3 not 2

there are 7 republicans, 10 democrats and 3 independents.

C(n,r) is the number of different combinations of n distinct objects taken r at a time.

C(n,r) =
`(n!)/((n-r)!r!)`

a) it will consist of 1 Republican and 4 Democrats

The number of ways this can be chosen is = C(7,1) × C(10,4) =
`(7!)/((7-1)!1!) * (10!)/((10-4)!4!)` = 7 × 210 = 1470 ways

(b) it will consist of 2 Republicans, 2 Democrats, and 2 Independents (consist of 2 independents not 3)

The number of ways this can be chosen is = C(7,2) × C(10,2) × C(2,2)

=
`(7!)/((7-2)!2!) * (10!)/((10-2)!2!)* (3!)/((3-2)!2!)` = 21 × 45 × 3 = 2835 ways