Problem 27 - College Algebra

Question

Evaluate each factorial expression. $$\frac{(n+2) !}{n !}$$

Step-by-Step Solution

Answer

The simplified form of \(\frac{(n+2) !}{n !}\) is \((n+2) \cdot (n+1)\)

Step 1: Expand the Factorial in the Numerator

The factorial operation expands a number into the product of all integers from one to that number. Expanding (n+2)! We can write it as \((n+2) \cdot (n+1) \cdot n!\)

Step 2: Simplify the Expression

The expression now becomes \(\frac{(n+2) \cdot (n+1) \cdot n!}{n !}\). Noticing that there is a \(n!\) both in the numerator and the denominator, these will cancel out.

Step 3: Final Simplification

After simplification, the final expression is \((n+2) \cdot (n+1)\). This is a simplified form of the given factorial expression.