Evaluate each factorial expression. 16!/2!14!

Verified step by step guidance

1. Recall the definition of factorial: for any positive integer \(n\), \(n! = n \times (n-1) \times (n-2) \times \cdots \times 1\).
2. Rewrite the given expression \(\frac{16!}{2! \times 14!}\) and observe that both numerator and denominator contain factorials that can be simplified.
3. Express \$16!\( as \(16 \times 15 \times 14!\) to allow cancellation with the \)14!$ in the denominator: \(\frac{16 \times 15 \times 14!}{2! \times 14!}\).
4. Cancel out the common \$14!$ terms in numerator and denominator, leaving \(\frac{16 \times 15}{2!}\).
5. Calculate \$2!$ which is \(2 \times 1 = 2\), then simplify the fraction \(\frac{16 \times 15}{2}\) by dividing the numerator by 2.

Key Concepts

Factorial Notation

Factorial notation, denoted by n!, represents the product of all positive integers from 1 up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. It is commonly used in permutations, combinations, and algebraic expressions.

Simplifying Factorial Expressions

When factorials appear in fractions, common terms can be canceled to simplify the expression. For instance, in 16!/2!14!, the 14! in the denominator cancels with part of 16! in the numerator, reducing the calculation to a simpler product.

Evaluating Factorial Expressions

After simplification, evaluate the remaining factorial terms by multiplying the integers. This step involves careful arithmetic to find the numerical value of the expression, ensuring accuracy in the final result.