Evaluate each factorial expression. 17!/15!

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. Write out the factorial expressions explicitly: \(17! = 17 \times 16 \times 15!\).
3. Substitute this into the given expression: \(\frac{17!}{15!} = \frac{17 \times 16 \times 15!}{15!}\).
4. Notice that \$15!$ appears in both numerator and denominator, so they cancel out: \(\frac{17 \times 16 \times \cancel{15!}}{\cancel{15!}} = 17 \times 16\).
5. The expression simplifies to the product \(17 \times 16\), which you can multiply to find the final value.

Key Concepts

Factorial Notation

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

Simplifying Factorial Expressions

When dividing factorials like 17!/15!, you can cancel common terms. Since 17! = 17 × 16 × 15!, the 15! terms cancel out, leaving 17 × 16. This simplification avoids calculating large factorials directly.

Properties of Factorials in Division

Factorials have the property that n! = n × (n-1)!. This allows breaking down factorial expressions in division problems to simplify calculations by canceling out common factorial terms, making complex expressions manageable.