Use the formula for _nP_r to evaluate each expression. ₈P₅

Verified step by step guidance

1. Recall the formula for permutations: \(nP_r = \frac{n!}{(n-r)!}\), where \(n\) is the total number of items and \(r\) is the number of items to arrange.
2. Identify the values of \(n\) and \(r\) from the problem: here, \(n = 8\) and \(r = 5\).
3. Substitute these values into the formula: \(8P_5 = \frac{8!}{(8-5)!} = \frac{8!}{3!}\).
4. Write out the factorial expressions explicitly: \(8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3!\), so \(\frac{8!}{3!} = \frac{8 \times 7 \times 6 \times 5 \times 4 \times 3!}{3!}\).
5. Cancel the common \$3!$ terms in numerator and denominator, leaving \(8P_5 = 8 \times 7 \times 6 \times 5 \times 4\), which you can multiply to find the final answer.

Key Concepts

Permutation (nPr)

A permutation represents the number of ways to arrange a subset of items from a larger set where order matters. The notation nPr denotes the number of permutations of n items taken r at a time.

Permutation Formula

The formula for permutations is nPr = n! / (n - r)!, where n! is the factorial of n. This formula calculates the total ordered arrangements of r items selected from n distinct items.

Factorial Function

The factorial of a positive integer n, written as n!, is the product of all positive integers from 1 to n. It is essential for computing permutations and combinations in algebra.