Enumerative Combinatorics · Counting principles, bijections, and the pigeonhole principle
Among n distinct volunteers, a specified set of r volunteers is qualified to serve as…
Problem
Among \(n\) distinct volunteers, a specified set of \(r\) volunteers is qualified to serve as spokesperson. A selection consists of an unordered committee of size \(k\) together with one of its qualified members designated spokesperson. Two choices with the same committee but different spokespeople count separately. For integers \(n\ge0\), \(0\le r\le n\), and \(0\le k\le n\), derive a formula for the number of selections by choosing the spokesperson first. Derive the same count by conditioning on the number \(j\) of qualified committee members. Use the convention \(\binom ab=0\) when \(a\ge0\) and \(b<0\) or \(b>a\). Handle the empty committee separately. Evaluate the count for \(n=8,r=3,k=4\).
Hint
A committee and a committee with a designated spokesperson are different counting objects.
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