HMMT-Feb 2010 guts, Problem 31

In the Democratic Republic of Irun, 5 people are voting in an election among 5 candidates. If each person votes for a single candidate at random, what is the expected number of candidates that will be voted for?

Solution: The probability that a chosen candidate will receive no votes at all is $\left(\frac{4}{5}\right)^{5}$, or the probability that every person will vote for someone other than that one candidate. Then the probability that a chosen candidate will receive at least one vote is $1-\left(\frac{4}{5}\right)^{5}=\frac{2101}{3125}$. By linearity of expectations, the final answer is this probability times the number of candidates, or $5 \cdot \frac{2101}{3125}=\frac{2101}{625}$. $\fbox{\frac{2101}{625}}$.

Answer: \frac{2101}{625}