Evaluate n!/(n-r)! for n = 20 and r = 3

Verified step by step guidance

1. Identify the given values: \( n = 20 \) and \( r = 3 \).
2. Recall the formula to evaluate: \( \frac{n!}{(n-r)!} \). This expression represents the number of permutations of \( n \) items taken \( r \) at a time.
3. Substitute the given values into the formula: \( \frac{20!}{(20-3)!} = \frac{20!}{17!} \).
4. Expand the factorial expressions to simplify the fraction. Since \( 20! = 20 \times 19 \times 18 \times 17! \), the \( 17! \) terms cancel out, leaving \( 20 \times 19 \times 18 \).
5. Multiply the remaining terms \( 20 \times 19 \times 18 \) to find the value of the expression.

Key Concepts

Factorial Notation

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. It is commonly used in permutations and combinations to count arrangements.

Permutation Formula

The expression n!/(n-r)! calculates the number of permutations of r objects chosen from n distinct objects. It counts the ordered arrangements and is essential for problems involving selection and ordering.

Substitution and Simplification

To evaluate n!/(n-r)!, substitute the given values for n and r, then simplify by canceling common factorial terms. This reduces the calculation to a product of consecutive integers, making it easier to compute.