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Three students are running a race. How many different ways can they come in first, second, and third?
Answer:

Three students are running a race. How many different ways can they come in first, second, and third? $\newline$ Answer:

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Experimental probability

Full solution

Q. Three students are running a race. How many different ways can they come in first, second, and third?

\newline

Answer:

Understand the Problem: Understand the problem. $\newline$ We need to find the number of different permutations of three students finishing a race in first, second, and third place. A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement. Since the order in which the students finish the race matters, we are dealing with permutations.
Calculate Permutations: Calculate the number of permutations. $\newline$ The number of permutations of $n$ distinct objects taken $r$ at a time is given by the formula: $\newline$ $P(n, r) = \frac{n!}{(n - r)!}$ $\newline$ In this case, we have $3$ students and we want to arrange all of them, so $n = r = 3$ .
Apply Formula: Apply the permutation formula. $\newline$ Using the formula from Step $2$ , we calculate: $\newline$ $P(3, 3) = \frac{3!}{(3 - 3)!}$ $\newline$ $P(3, 3) = \frac{3!}{0!}$ $\newline$ We know that $3!$ ( $3$ factorial) is $3 \times 2 \times 1 = 6$ and $0!$ ( $0$ factorial) is defined to be $1$ . $\newline$ So, $P(3, 3) = \frac{6}{1}$ $\newline$ $P(3, 3) = 6$
Conclude Solution: Conclude the solution. $\newline$ There are $6$ different ways the three students can come in first, second, and third in the race.

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