In Exercises 1–8, use the formula for nPr to evaluate each expression. ₆P₆

Verified step by step guidance

1. Recall the formula for permutations: $\mathrm{nP}r=\; \backslash frac\{n!\}\{(n-r)!\}$, where $n$ is the total number of items and $r$ is the number of items to arrange.
2. Identify the values of $n$ and $r$ in the expression $6P6$. Here, $n\; =\; 6$ and $r\; =\; 6$.
3. Substitute the values into the formula: $6P6\; =\; \backslash frac\{6!\}\{(6-6)!\}\; =\; \backslash frac\{6!\}\{0!\}$.
4. Recall that $0!\; =\; 1$ by definition, so the expression simplifies to $6P6\; =\; \backslash frac\{6!\}\{1\}\; =\; 6!$.
5. Evaluate $6!$ by multiplying all positive integers from 6 down to 1: $6!\; =\; 6\; \backslash times\; 5\; \backslash times\; 4\; \backslash times\; 3\; \backslash times\; 2\; \backslash times\; 1$.

Key Concepts

Permutation (nPr)

A permutation represents the number of ways to arrange 'r' objects from a set of 'n' distinct objects, where order matters. The formula for permutations is nPr = n! / (n - r)!, which calculates all possible ordered arrangements.

Factorial Notation

Factorial, denoted by '!', is the product of all positive integers up to a given number. For example, 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720. Factorials are essential in permutation calculations to count arrangements.

Evaluating Permutations When n = r

When the number of objects chosen equals the total number available (n = r), the permutation formula simplifies to nPr = n!, since (n - r)! = 0! = 1. This means arranging all objects in every possible order.