Evaluate each factorial expression. 18!/16!

Verified step by step guidance

1. Recall the definition of factorial: for a positive integer \(n\), \(n! = n \times (n-1) \times (n-2) \times \cdots \times 1\).
2. Write out the factorial expressions explicitly: \(18! = 18 \times 17 \times 16!\).
3. Substitute \$18!$ in the expression \(\frac{18!}{16!}\) with \(18 \times 17 \times 16!\) to get \(\frac{18 \times 17 \times 16!}{16!}\).
4. Cancel the common factor \$16!$ in the numerator and denominator, leaving \(18 \times 17\).
5. Multiply the remaining numbers \(18\) and \(17\) to find the value of the expression.

Key Concepts

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.