Question

How many ways are there to select a committee of 17 politicians chosen from a room full of indistinguishable Democrats, indistinguishable Republicans, and indistinguishable Independents if every party must have at least two members on the committee? If, in addition, no group may have a majority of the committee members?

Answer 1

Answer:

a) 78

b) 33

Explanation:

In order to have majority, the number of members is one party must be greater than the number of members in other party.

This means that  one of the party must have at least 9 members.

a) Every party must have at least two members on the committee

$\frac{(n+r-1)!}{r!} \\\frac{11+3-1}{11} \\=^{13}C_{11}\\= \frac{13 *12 *11!}{11! *2} \\= 13 *6\\= 78$

b) no group may have a majority of the committee members

Since , both the parties cannot have 9 members thus in order to compute the committee number in a way  that no group may have a majority of the committee members

78 - ^{10} C_8

78-45

33