Simplifying expression of mean value of number of hearts when drawing 5 from a deck

Question

Draw 5 cards from a deck of 32 cards (8 of each suit). $X$ counts the number of hearts.

For $k\in\lbrace 0,...,5\rbrace $ I have $P(X=k)=\frac{C_8^k C_{24}^{5-k}}{C_{32}^5}$.

I'm supposed to "calculate" $E(X)$. I really can't see what more I can do than write it from the very definition : $E(X)=\Sigma_{k=0}^5k\frac{C_8^k C_{24}^{5-k}}{C_{32}^5}$

Does this turn out to have a simpler expression ?

Answer 1

The expected number of hearts must equal the expected number of any other suit.

The sum of the expected numbers for all 4 suits must equal the number of cards chosen i.e. 5.

The expected number of hearts therefore equals $\frac{5}{4}.$

Answer 2

This is a hypergeometric distribution, whose expected value (mean) is: $$\mathbb E(X)=n\cdot \frac{K}{N}=5\cdot \frac8{32}=1.25.$$ WA answer for the sum: $$\mathbb E(X)=\sum_{k=0}^5 k\cdot \frac{{8\choose k}{24\choose 5-k}}{{32\choose 5}}=1.25.$$