Use a calculator's factorial key to evaluate each expression. 20!/(20−3)!

Verified step by step guidance

1. Identify the expression given: \(\frac{20!}{(20-3)!}\).
2. Simplify the denominator inside the factorial: calculate \$20 - 3\( to get \)17\(, so the expression becomes \)\frac{20!}{17!}$.
3. Recall the definition of factorial: \(n! = n \times (n-1) \times (n-2) \times \cdots \times 1\).
4. Express \$20!\( in terms of \)17!\( to simplify the fraction: \)20! = 20 \times 19 \times 18 \times 17!$.
5. Cancel out \$17!\( in numerator and denominator, leaving \)20 \times 19 \times 18$, which you can then multiply using a calculator.

Key Concepts

Factorials

A factorial, denoted 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 commonly used in permutations, combinations, and probability calculations.

Permutation Formula

The expression 20! / (20−3)! represents the number of permutations of 20 items taken 3 at a time. It counts the ordered arrangements of 3 elements selected from 20, calculated by dividing the factorial of the total items by the factorial of the difference.

Using a Calculator's Factorial Function

Many scientific calculators have a factorial key (!) to compute factorial values quickly. To evaluate expressions like 20!/(20−3)!, you calculate 20! and 17! separately or use the permutation function if available, simplifying the process and reducing manual errors.