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Seven runners are competing in a race where $3$ of them will earn medals for finishing first, second, and third. How many unique ways are there to arrange $3$ of the $7$ runners in first, second, and third place?

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Compound events: find the number of outcomes

Full solution

Q. Seven runners are competing in a race where

3

of them will earn medals for finishing first, second, and third. How many unique ways are there to arrange

3

of the

7

runners in first, second, and third place?

Calculate Factorial of $7$ : We need to calculate the number of permutations of $7$ runners taken $3$ at a time, since the order in which they finish is important. $\newline$ The formula for permutations of $n$ items taken $r$ at a time is $P(n, r) = \frac{n!}{(n - r)!}$ . $\newline$ Here, $n = 7$ (total runners) and $r = 3$ (positions to fill).
Calculate Factorial of $(7-3)$ : First, we calculate the factorial of $n$ , which is $7!$ ( $7$ factorial). $\newline$ $7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040$ .
Use Permutation Formula: Next, we calculate the factorial of $(n - r)$ , which is $(7 - 3)!$ or $4!$ ( $4$ factorial). $4! = 4 \times 3 \times 2 \times 1 = 24$ .
Perform Division: Now, we use the permutation formula to find $P(7, 3)$ . $P(7, 3) = \frac{7!}{(7 - 3)!} = \frac{5040}{24}$ .
Perform Division: Now, we use the permutation formula to find $P(7, 3)$ . $P(7, 3) = \frac{7!}{(7 - 3)!} = \frac{5040}{24}$ .We perform the division to find the number of unique ways. $P(7, 3) = \frac{5040}{24} = 210$ .

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