In Exercises 81–85, use a calculator's factorial key to evaluate each expression. 20!/(20−3)!

Verified step by step guidance

1. Identify the expression to evaluate: $\frac{20!}{(20-3)!}$.
2. Simplify the expression inside the factorial in the denominator: $(20-3)!=17!$.
3. Rewrite the expression as $\frac{20!}{17!}$.
4. Recognize that $20!=20\times 19\times 18\times 17!$.
5. Cancel out $17!$ in the numerator and denominator, leaving $20\times 19\times 18$.

Key Concepts

Factorial

A factorial, denoted as n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are commonly used in permutations and combinations, where they help calculate the number of ways to arrange or select items.

Permutations

Permutations refer to the different ways of arranging a set of items where the order matters. The formula for permutations of n items taken r at a time is given by n!/(n-r)!. In the context of the question, evaluating 20!/(20-3)! calculates the number of ways to arrange 3 items from a set of 20.

Calculator Functions

Most scientific calculators have a factorial function, often labeled as 'n!'. This function allows users to compute factorials quickly without manual multiplication. Understanding how to use this function is essential for efficiently solving problems involving large factorials, as they grow rapidly in size.