Problem 78 - College Algebra

Question

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

Step-by-Step Solution

Answer

By using the property of factorials, \(\frac{900 !}{899 !}\) simplifies to 900.

Step 1: Understand Factorial Notation

Factorial notation often presented as (n!). It means the product of all positive integers from 1 to n. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.

Step 2: Applying Factorial Property

The next step is to use the property of factorials where the factorial of a number n (n!) is equal to n times the factorial of (n-1). As so, \( n! = n × (n-1)! \). Whenever this rule applies, you can simplify the expression.

Step 3: Simplify The Expression

Applying the factorial property, 900! can also be written as (900 x 899!) So, the expression becomes \(\frac{900 x 899 !}{899 !}\). As you can see, 899! is common in the numerator and the denominator. So, it cancels out.

Step 4: Final Calculation

After the 899! cancels out, the final expression becomes 900. Hence, \(\frac{900 !}{899 !}\) = 900 without any computation.