In Exercises 1–8, use the formula for nPr to evaluate each expression. 9P4

Verified step by step guidance

1. Recall the formula for permutations: $\mathrm{nP}r=\frac{n!}{(n-r)!}$, 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: $9P4\; =\frac{9!}{(9\; -\; 4)!}=\frac{}{}$.
4. Write out the factorial expressions explicitly to simplify: $9!\; =\; 9\; imes\; 8\; imes\; 7\; imes\; 6\; imes\; 5!$, so the $5!$ terms will cancel out.
5. Simplify the expression by canceling $5!$ and multiplying the remaining terms: $9\; imes\; 8\; imes\; 7\; imes\; 6$. This product gives the number of permutations.

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

Factorial, denoted by n!, is the product of all positive integers from 1 to n. It is essential in permutations to compute the number of ways to arrange items, for example, 5! = 5 × 4 × 3 × 2 × 1 = 120.