Problem 81 - College Algebra

Question

Use a calculator's factorial key to evaluate each expression. $$\frac{200 !}{198 !}$$

Step-by-Step Solution

Answer

The simplified expression equals \( 39800 \).

Step 1: Understand Factorials

Factorials are the product of a number and all the whole numbers below it down to 1. So \(200!\) means \(200 \times 199 \times 198 \times 197 \times ... \times 3 \times 2 \times 1\). Similarly, \(198!\) means \(198 \times 197 \times ... \times 3 \times 2 \times 1\). So the given fraction can be simplified by dividing out the common terms in the numerator and the denominator.

Step 2: Simplify the Expression

The expression \( \frac{200 !}{198 !} \) can be simplified as \( \frac{200 \times 199 \times 198!}{198!} \). Since \(198!\) is a common term in both the numerator and the denominator, it cancels out, leaving: \(200 \times 199\).

Step 3: Evaluate the Answer

Finally, we just have to calculate the product of 200 and 199 to get the answer.