Evaluate $\frac{40!}{4!38!}$

Verified step by step guidance

1. Recognize that the expression \(\frac{40!}{4! \times 38!}\) is a combination formula, which can be written as \(\binom{40}{4}\), representing the number of ways to choose 4 items from 40.
2. Recall the combination formula: \(\binom{n}{r} = \frac{n!}{r! (n-r)!}\), where \(n=40\) and \(r=4\) in this problem.
3. Rewrite the expression using the formula: \(\binom{40}{4} = \frac{40!}{4! (40-4)!} = \frac{40!}{4! \times 36!}\).
4. Notice that the original problem has \$38!\( in the denominator instead of \)36!$, so carefully check the factorial terms to confirm the expression matches the combination formula or simplify accordingly.
5. If simplifying directly, expand the factorials to cancel common terms: write \$40!\( as \)40 \times 39 \times 38!\(, then cancel \)38!\( in numerator and denominator, and simplify the remaining expression by dividing by \)4!$.

Key Concepts

Factorials

A factorial, denoted by n!, is the product of all positive integers from 1 up to n. For example, 4! = 4 × 3 × 2 × 1 = 24. Factorials are commonly used in permutations, combinations, and algebraic expressions involving sequences.

Simplifying Factorial Expressions

When factorials appear in fractions, common terms can often be canceled to simplify the expression. For instance, in 40! / (4! 38!), the 38! in the denominator cancels with part of 40!, reducing the complexity of the calculation.

Binomial Coefficients

The expression 40! / (4! 38!) represents a binomial coefficient, often read as '40 choose 4'. It counts the number of ways to select 4 items from 40 without regard to order, and is fundamental in combinatorics and probability.