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In a race with 17 horses, how many different trifecta bets can be made. a trifecta is when you pick the first three finishers in the exact order.

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3 Answers

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$17\times 16\times 15$. There are $17$ choices for first. For each of these there are $16$ choices for second. For each of these combinations there are $15$ choices for third.

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Here is one way to look at the problem:

1. Choose $3$ out of $17$ horses: $\binom{17}{3}=\frac{17!}{14!\times3!}=680$
2. Multiply by the number of different ways to order these $3$ horses: $3!=6$

And the result is $680\times6=4080$.

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Also, since order matters this is a permutation:

$$ P_{3}^{17} = \frac{17!}{14!} = 17 * 16 * 15 = 4080 $$

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