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There were 18 students running in a race. How many different arrangements of first, second, and third place are possible?
Answer:

There were $18$ students running in a race. How many different arrangements of first, second, and third place are possible? $\newline$ Answer:

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Experimental probability

Full solution

Q. There were

18

students running in a race. How many different arrangements of first, second, and third place are possible?

\newline

Answer:

Calculate Permutations Formula: To determine the number of different arrangements for the first three places, we need to calculate the permutations of $18$ students taken $3$ at a time. The formula for permutations is $P(n, k) = \frac{n!}{(n - k)!}$ , where $n$ is the total number of items and $k$ is the number of items to arrange.
Calculate Factorial of $18$ : First, we calculate the factorial of $18$ , which is $18! = 18 \times 17 \times 16 \times \ldots \times 1$ .
Calculate Factorial of Difference: Next, we calculate the factorial of the difference between the total number of students and the number of places, which is $(18 - 3)! = 15! = 15 \times 14 \times \ldots \times 1$ .
Use Permutation Formula: Now, we use the permutation formula $P(18, 3) = \frac{18!}{(18 - 3)!} = \frac{18!}{15!}$ .
Simplify Expression: We simplify the expression by canceling out the common factorial terms: $P(18, 3) = \frac{18 \times 17 \times 16}{1} = 18 \times 17 \times 16$ .
Perform Multiplication: Finally, we perform the multiplication: $18 \times 17 \times 16 = 4896$ .

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