In Exercises 41-44, perform the indicated calculation.
42. 8P6

Verified step by step guidance

1. Step 1: Understand the notation 8P6. This represents a permutation, which is the number of ways to arrange 6 items out of 8 in a specific order. The formula for permutations is given by P(n, r) = n! / (n - r)!, where n is the total number of items, and r is the number of items to arrange.
2. Step 2: Identify the values of n and r in the problem. Here, n = 8 and r = 6.
3. Step 3: Substitute the values of n and r into the permutation formula. This gives P(8, 6) = 8! / (8 - 6)!. Simplify the denominator to get P(8, 6) = 8! / 2!.
4. Step 4: Expand the factorials in the numerator and denominator. Recall that 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 and 2! = 2 × 1. Cancel out the common terms in the numerator and denominator.
5. Step 5: Multiply the remaining terms in the numerator after cancellation to compute the result. This will give you the total number of permutations for arranging 6 items out of 8.

Key Concepts

Permutations

Permutations refer to the different ways of arranging a set of items where the order matters. The notation nPr represents the number of ways to choose r items from a total of n items, considering the arrangement. For example, if you have 3 letters A, B, and C, the permutations of choosing 2 letters would include AB, AC, BA, BC, CA, and CB.

Factorial

A factorial, denoted as n!, is the product of all positive integers up to n. It is a fundamental concept in combinatorics, used to calculate permutations and combinations. For instance, 5! equals 5 × 4 × 3 × 2 × 1 = 120. Factorials grow rapidly, making them essential for counting arrangements and selections.

Calculation of Permutations

The formula for calculating permutations is given by nPr = n! / (n - r)!. This formula allows you to determine the number of ways to arrange r items from a total of n items. In the case of 8P6, you would calculate it as 8! / (8 - 6)! = 8! / 2!, which simplifies the computation by reducing the factorial terms.