Problem 78 - Algebra and Trigonometry

Question

Explain the best way to evaluate \(\frac{900 !}{899 !}\) without a calculator.

Step-by-Step Solution

Answer

The evaluation of the expression \(\frac{900 !}{899 !}\) gives the result as \(900\).

Step 1: Understand factorial operation

For any natural number n, the factorial operation (\(n!\)) gives the product of all positive integers less than or equal to n. That is, \(n! = n×(n−1)×…×3×2×1\).

Step 2: Apply the property of factorials

The property of factorials presents that \(n!=n×(n−1)!\). Thus, according to this property, \(900!=900×899!\).

Step 3: Simplify the expression

Substitute \(900!\) with \(900×899!\) in the given expression. Thus, \(\frac{900 !}{899 !}= \frac{900×899 !}{899 !}\). Since \(899 !\) is common in the numerator and denominator, they can be cancelled out. The final result is then \(900\).