Evaluate each factorial expression. 17!/15!

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Recall the definition of factorial: for any positive integer $n$, $n! = n \times (n-1) \times (n-2) \times \cdots \times 1$.

Write out the factorial expressions explicitly: $17! = 17 \times 16 \times 15!$.

Substitute this into the given expression: $\frac{17!}{15!} = \frac{17 \times 16 \times 15!}{15!}$.

Notice that \$15!$ appears in both numerator and denominator, so they cancel out: $\frac{17 \times 16 \times \cancel{15!}}{\cancel{15!}} = 17 \times 16$.

The expression simplifies to the product $17 \times 16$, which you can multiply to find the final value.

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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.