In Exercises 23–28, evaluate each factorial expression. (n+2)!/n!

Verified step by step guidance

1. Identify the given expression: \( \frac{(n+2)!}{n!} \).
2. Recall the definition of a factorial: \( n! = n \times (n-1) \times (n-2) \times \ldots \times 1 \).
3. Expand \((n+2)!\) using the definition of factorial: \((n+2)! = (n+2) \times (n+1) \times n!\).
4. Substitute the expanded form of \((n+2)!\) into the expression: \( \frac{(n+2) \times (n+1) \times n!}{n!} \).
5. Cancel out \(n!\) from the numerator and the denominator, resulting in \((n+2) \times (n+1)\).

Key Concepts

Factorial Definition

A factorial, denoted as n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are used in permutations, combinations, and various mathematical calculations, making them fundamental in algebra.

Properties of Factorials

Factorials have specific properties that simplify calculations. One important property is that (n + 1)! = (n + 1) × n!. This property allows us to express factorials in terms of smaller factorials, which is crucial for simplifying expressions like (n + 2)!/n!.

Simplifying Factorial Expressions

To evaluate expressions involving factorials, we can often cancel terms. For instance, in (n + 2)!/n!, we can expand (n + 2)! as (n + 2)(n + 1)n!, allowing us to simplify the expression to (n + 2)(n + 1), which is easier to compute.