Chapter 1, Problem 1.10TE

From a group of n people, suppose that we want to choose a committee of $k,k\le n$, one of whom is to be designated as chairperson.

a. By focusing first on the choice of the committee and then on the choice of the chair, argue that there are $\left(\begin{array}{c}n\\ k\end{array}\right)k$ possible choices.

b. By focusing first on the choice of the nonchair committee members and then on the choice of the chair, argue that there are $\left(\begin{array}{c}n\\ k-1\end{array}\right)\left(n-k+1\right)$ possible choices.

c. By focusing first on the choice of the chair and then on the choice of the other committee members, argue that there are $\left(\begin{array}{c}n-1\\ k-1\end{array}\right)$ possible choices.

d. Conclude from parts (a), (b), and (C) that $k\left(\begin{array}{c}n\\ k\end{array}\right)=\left(n-k+1\right)\left(\begin{array}{c}n\\ k-1\end{array}\right)=n\left(\begin{array}{c}n-1\\ k-1\end{array}\right)$

e. Use the factorial definition of $\left(\begin{array}{c}m\\ r\end{array}\right)$ to verify the identity in part (d).