Suppose we have a committee of 11 people, 1 of which is the chairman, and the remaining 10 can be one of the 3 groups: A, B and C (where there are no restrictions on how many each group can contain as long as the total number of people are 10).
I am trying to find 3 things:
- The number of different configurations for a full committee: would this just be the solution to $c_1$ + $c_2$ + $c_3$ = $10$ ?
- The number of configurations for a full committee if each of the 3 groups contains at least 2 members? would this be the solution to $c_1$ + $c_2$ + $c_3$ = $6$ ?
- The final one I am struggling with is: how many ways can we assigns the 11 people to different positions in the committee if there has to be one chairman and no restriction on the remaining groups? For this one, there are 11 ways to choose a chairman and then ${10}\choose{4}$ for the remaining groups? Or is there a different way to approach this question?
Much appreciated!
The first is correct if by configurations you mean the number of people in each group. You should specify that $0$ is an allowable value for the variables, then you can use stars and bars to find the answer. For the second, you have already accounted for six people, so the sum should be $4$. For the third, I read it to make the people distinguishable now, so each person can be on any of three groups, giving $11\cdot 3^{10}$ where the $11$ chooses the chairman.