A ballot contains a list of $5$ candidates. Each voter can choose $0$ to $5$ candidates. In how many ways can a voter complete the ballot?
$17$
$20$
$32$
$125$
I tried doing $5!$ and I get $120$...
2026-04-02 05:31:36.1775107896
How many ways to vote?
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If choices are not ordered as @Aston suggests, then any of the five candidates can be either chosen or not: $2^5 = 32.$ Also, $\sum_{i=0}^5 {5 \choose i} = 32.$
Note: By the binomial theorem, $(x + y)^5 = \sum_{i=0}^5 {5 \choose i}x^iy^{5-i}.$ Take $x = y = 1$ to show that the two methods count the same outcomes.