Simplify k! / (k + 2)! for any integer k ≥ 0.

Verified step by step guidance

1. Recall the definition of factorial: for any integer n ≥ 0, n! = n × (n - 1) × (n - 2) × ... × 1, and 0! = 1 by convention.
2. Express the denominator (k + 2)! in terms of k!: write (k + 2)! as (k + 2) × (k + 1) × k!.
3. Rewrite the original expression \( \frac{k!}{(k + 2)!} \) by substituting the expanded form of (k + 2)!: \( \frac{k!}{(k + 2) \times (k + 1) \times k!} \).
4. Cancel the common factor k! in numerator and denominator, simplifying the expression to \( \frac{1}{(k + 2) \times (k + 1)} \).
5. Conclude that the simplified form of \( \frac{k!}{(k + 2)!} \) is \( \frac{1}{(k + 2)(k + 1)} \) for any integer k ≥ 0.

Key Concepts

Factorial Definition

The factorial of a non-negative integer n, denoted n!, is the product of all positive integers from 1 up to n. By definition, 0! = 1. Understanding factorials is essential for simplifying expressions involving factorial terms.

Factorial Expansion and Cancellation

To simplify ratios of factorials like k! / (k + 2)!, expand the larger factorial to reveal common factors. For example, (k + 2)! = (k + 2)(k + 1)k!, allowing cancellation of k! in numerator and denominator.

Simplification of Rational Expressions

After canceling common factorial terms, simplify the remaining expression by performing arithmetic operations. This often reduces complex factorial ratios to simpler algebraic fractions or products.