Use the formula for _nP_r to evaluate each expression. ₁₀P₄

Verified step by step guidance

1. Recall the formula for permutations: \(\displaystyle {}_nP_r = \frac{n!}{(n-r)!}\), 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, \(n = 10\) and \(r = 4\).
3. Substitute these values into the permutation formula: \(\displaystyle {}_{10}P_4 = \frac{10!}{(10-4)!} = \frac{10!}{6!}\).
4. Simplify the factorial expression by expanding \$10!\( only as far as needed to cancel with \)6!\(: \)\displaystyle \frac{10 \times 9 \times 8 \times 7 \times 6!}{6!}$.
5. Cancel the \$6!\( terms and multiply the remaining numbers: \)10 \times 9 \times 8 \times 7$ to find the number of permutations.

Key Concepts

Permutation Formula

The permutation formula, denoted as nPr, calculates the number of ways to arrange r objects from a set of n distinct objects where order matters. It is given by nPr = n! / (n - r)!, where '!' denotes factorial.

Factorial Function

A factorial, represented by n!, is the product of all positive integers from 1 up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are essential in permutations and combinations to count arrangements.

Evaluating Permutations

To evaluate a permutation like 10P4, substitute into the formula: 10P4 = 10! / (10 - 4)! = 10! / 6!. Simplify by canceling common factorial terms and compute the product of the remaining factors.