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.