The following is from: http://www.math.sjsu.edu/~bremer/Teaching/Math163/Homework/HomeworkFiles/Solution03.pdf
I am having trouble understanding these identities and the solutions. I am confused as to the LHS because I thought the number of ways to choose a $k$ person committee with a chairperson from a group of $n$ people is $k \cdot \binom{n}{k}$.
This solution seems likes its counting the same thing but giving another answer. Why the summation?
Both sides count committees of all sizes. On the right-hand side, the arbitrary size arises because $2^{n-1}$ allows every non-chairperson to be in the committee or not independently. On the left-hand side, we form chaired committees of $k$ people, so we have to sum over $k$ to count all committees of all sizes. You're right in thinking that $k\binom nk$ counts the number of ways to choose a $k$-person committee with a chairperson from a group of $n$ people.
