Evaluate 40!/(4! 38!)

Verified step by step guidance

1. Recognize that the expression \( \frac{40!}{4! \times 38!} \) is a combination formula, specifically \( \binom{40}{4} \).
2. Recall the combination formula: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \).
3. Identify \( n = 40 \) and \( r = 4 \) in the formula.
4. Substitute the values into the formula: \( \binom{40}{4} = \frac{40!}{4!(40-4)!} \).
5. Simplify the expression by canceling \( 38! \) in the numerator and denominator, leaving \( \frac{40 \times 39 \times 38!}{4! \times 38!} \), and then simplify further to find the result.

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 for counting problems in algebra and probability.

Binomial Coefficient

The expression n!/(k!(n-k)!) represents the binomial coefficient, often read as 'n choose k'. It counts the number of ways to choose k elements from a set of n elements without regard to the order of selection. This concept is fundamental in combinatorics and is used in various algebraic applications.

Simplification of Fractions

Simplifying fractions involves reducing them to their lowest terms by canceling common factors in the numerator and denominator. In the context of factorials, this means recognizing and eliminating terms that appear in both the numerator and denominator, which can significantly ease calculations.