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?
I was able to find the number of ways to select a committee if each party must have at least two members on the committee: $\binom{11+2}{2}=\binom{13}{2}=78$ ways.
I am confused on the second part of the problem.
Any help is appreciated.
Best Answer
To have a majority means that one party has at least $9$ members. As it is impossible for two parties to each have $9$ members, you can compute the number of ways to form a committee with at least $9$ Democrats and at least two of each other party, multiply by $3$, and subtract from your $78$ ways without the restriction.