In Exercises 23–28, evaluate each factorial expression. 20!/2!18!

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Identify the factorial expression: \( \frac{20!}{2!18!} \).

Recognize that \( n! \) (n factorial) is the product of all positive integers up to \( n \).

Notice that the expression can be simplified by canceling common terms in the numerator and denominator.

Rewrite the expression: \( \frac{20 \times 19 \times 18!}{2! \times 18!} \).

Cancel \( 18! \) from the numerator and denominator, then simplify the remaining expression: \( \frac{20 \times 19}{2!} \).

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Factorial Definition

A factorial, denoted by 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, combinations, and various mathematical calculations, making them fundamental in algebra.

Binomial Coefficient

The expression n!/(k!(n-k)!) represents the binomial coefficient, which counts the number of ways to choose k elements from a set of n elements without regard to the order of selection. This is crucial in combinatorics and is often written as C(n, k) or 'n choose k'.

Simplifying Factorials

When evaluating expressions involving factorials, simplification is key. For instance, in the expression 20!/2!18!, the 20! can be rewritten as 20 × 19 × 18!, allowing for cancellation of the 18! in the numerator and denominator, simplifying the calculation significantly.