Evaluate each factorial expression. 18!/16!

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

Write out the factorial expressions explicitly: $18! = 18 \times 17 \times 16!$.

Substitute \$18!$ in the expression $\frac{18!}{16!}$ with $18 \times 17 \times 16!$ to get $\frac{18 \times 17 \times 16!}{16!}$.

Cancel the common factor \$16!$ in the numerator and denominator, leaving $18 \times 17$.

Multiply the remaining numbers $18$ and $17$ to find the value of the expression.

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Factorial Notation

Factorial notation, denoted by an exclamation mark (!), represents the product of all positive integers up to a given number. For example, n! = n × (n-1) × ... × 2 × 1. It is commonly used in permutations, combinations, and algebraic expressions.

Simplifying Factorial Expressions

When dividing factorials like 18!/16!, common terms can be canceled out. Since 18! = 18 × 17 × 16!, the expression 18!/16! simplifies to 18 × 17. This method avoids calculating large factorial values directly.

Properties of Factorials in Division

Factorials have a recursive property where n! = n × (n-1)!. This allows breaking down factorial expressions in division to simplify calculations by canceling out the smaller factorial, making complex expressions manageable.