In Exercises 81–85, use a calculator's factorial key to evaluate each expression.
$\frac{54!}{(54-3)!3!}$

Verified step by step guidance

1. Recognize that the expression $\frac{{54}^{!}}{{(54-3)}^{!}3!}$ is a combination formula in disguise, specifically $C(n,\; r)\; =\; \backslash frac\{n!\}\{(n-r)!r!\}$ where $n=54$ and $r=3$.
2. Rewrite the expression as $C(54,\; 3)\; =\; \backslash frac\{54!\}\{(54-3)!3!\}\; =\; \backslash frac\{54!\}\{51!3!\}$ to clearly see the combination structure.
3. Understand that factorial notation means the product of all positive integers up to that number, for example, $54!\; =\; 54\; \backslash times\; 53\; \backslash times\; 52\; \backslash times\; 51!.$ This allows simplification by canceling $51!$ in numerator and denominator.
4. Simplify the fraction by canceling $51!$ from numerator and denominator, leaving $\backslash frac\{54\; \backslash times\; 53\; \backslash times\; 52\}\{3!\}$.
5. Calculate $3!\; =\; 3\; \backslash times\; 2\; \backslash times\; 1\; =\; 6$, then divide the product $54\; \backslash times\; 53\; \backslash times\; 52$ by 6 to find the value of the expression.

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

Permutations count the number of ways to arrange r objects from a set of n distinct objects, given by n! / (n - r)!. This formula accounts for order and is essential for problems involving arrangements or sequences.

Using a Calculator for Factorials

Many scientific calculators have a factorial function (n!) that allows quick computation of large factorials. This is useful for evaluating expressions like 54! / (54−3)! 3! without manual multiplication.