In Exercises 81–85, use a calculator's factorial key to evaluate each expression. 54!/(54−3)!3!

Verified step by step guidance

1. Identify the expression to evaluate: $\frac{54!}{(54-3)!3!}$.
2. Simplify the expression inside the factorial: $(54-3)!=51!$.
3. Rewrite the expression using the simplified factorial: $\frac{54!}{51!\cdot 3!}$.
4. Recognize that $\frac{54!}{51!}$ simplifies to $54\times 53\times 52$ because the terms from 51! cancel out.
5. Calculate $3!$ which is $3\times 2\times 1$.

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 used in permutations and combinations, making them essential in probability and statistics.

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)!. This concept is crucial for solving problems involving arrangements and selections.

Combinations

Combinations are selections of items where the order does not matter. The formula for combinations of n items taken r at a time is n!/(r!(n-r)!). Understanding combinations is important for calculating probabilities and making selections from a larger set.