jmc

Question

At the end of the year, the Math Club decided to hold an election for which 5 equal officer positions were available. However, 16 candidates were nominated, of whom 7 were past officers. Of all possible elections of the officers, how many will have at least 1 of the past officers?

Accepted Answer

The number of total ways to choose the 5 officers is 165 = 4368. Of these, the number of ways to choose the officers without ANY of the past officers is 95 = 126. Thus, the number of ways to choose the 5 officers with at least 1 past officer is 4368 - 126 = 4242.