In Exercises 23–28, evaluate each factorial expression. 16!/2!14!

Verified step by step guidance

1. Identify the factorial expression: \( \frac{16!}{2!14!} \).
2. Recall the definition of a factorial: \( n! = n \times (n-1) \times (n-2) \times \ldots \times 1 \).
3. Notice that \( 16! = 16 \times 15 \times 14! \).
4. Substitute \( 16! \) in the expression: \( \frac{16 \times 15 \times 14!}{2! \times 14!} \).
5. Cancel out \( 14! \) from the numerator and the denominator, then simplify the remaining expression: \( \frac{16 \times 15}{2!} \).

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, combinations, and various mathematical calculations, making them fundamental in algebra and probability.

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 concept is crucial in combinatorics and is often used in problems involving combinations.

Simplifying Factorials

When evaluating expressions involving factorials, simplification is key. For instance, in the expression 16!/2!14!, the factorials can be simplified by canceling common terms. This process helps in reducing complex calculations and is essential for efficiently solving factorial expressions.