Use the formula for _nP_r to evaluate each expression. ₉P₄

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, \(n = 9\) and \(r = 4\).
3. Substitute these values into the formula: \(9P4 = \frac{9!}{(9-4)!} = \frac{9!}{5!}\).
4. Write out the factorial expressions to simplify: \(9! = 9 \times 8 \times 7 \times 6 \times 5!\), so \(\frac{9!}{5!} = \frac{9 \times 8 \times 7 \times 6 \times 5!}{5!}\).
5. Cancel the common \$5!$ terms to get \(9P4 = 9 \times 8 \times 7 \times 6\), which you can then multiply to find the final value.

Key Concepts

Permutation (nPr)

A permutation refers to the arrangement of objects in a specific order. The notation nPr represents the number of ways to arrange r objects out of n distinct objects, where order matters.

Permutation Formula

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

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 in permutation calculations to determine the number of possible arrangements.