In Exercises 1–8, use the formula for nPr to evaluate each expression. ₈P₀

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 chosen.
2. Identify the values of n and r from the problem: here, n = 8 and r = 0.
3. Substitute the values into the formula: $8P0\; =\frac{8!}{(8\; -\; 0)!}=\frac{}{}8!$.
4. Simplify the factorial expressions: since $8!\; /\; 8!\; =\; 1$, this step shows the value of the permutation.
5. Conclude that $8P0$ equals 1, because there is exactly one way to arrange zero items from eight.

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 and Special Cases

Factorial (n!) is the product of all positive integers up to n. By definition, 0! = 1. This is important for cases like 8P0, where r = 0, resulting in 8P0 = 8! / 8! = 1, meaning there is exactly one way to arrange zero items.