By Contributor • Updated Aug 30, 2022

Factorials, denoted by the exclamation mark, are a foundational concept in combinatorics and probability. They represent the product of all positive integers up to a given number. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Because factorials grow rapidly, dividing two of them might seem daunting at first. However, a simple algebraic shortcut can reduce the calculation to a few basic multiplications.

Step 1 – Write the Division as a Fraction

Express the two factorials you wish to divide in a fraction. For instance, to divide 11! by 8!, write 11! ÷ 8!.

Step 2 – Identify the Larger Factorial

Determine which factorial is larger. In this case, 11! is greater than 8! because 11 > 8.

Step 3 – Expand the Larger Factorial Until the Smaller One Appears

Rewrite the larger factorial so that the smaller one is a factor of it: 11! = 11 × 10 × 9 × 8!.

Step 4 – Cancel Common Factors

Divide numerator and denominator by the common factor 8!: (11 × 10 × 9 × 8!) ÷ 8! = 11 × 10 × 9.

Step 5 – Perform the Remaining Multiplication

Compute the product: 11 × 10 × 9 = 990. Thus, 11! ÷ 8! = 990.

What You’ll Need

• Paper
• Pencil or pen

This method works for any pair of factorials where the numerator’s n is larger than the denominator’s k (n ≥ k). It eliminates the need to calculate large factorials directly, saving time and reducing computational errors.