Edge Cases Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Check all edge cases for the statement: 'For natural numbers nn, n!(n−1)!=n\dfrac{n!}{(n-1)!} = n.' Test n=0n = 0 and n=1n = 1.

Solution

  1. 1
    For n≥2n \ge 2: n!(n−1)!=n⋅(n−1)!(n−1)!=n\frac{n!}{(n-1)!} = \frac{n \cdot (n-1)!}{(n-1)!} = n. Formula works.
  2. 2
    For n=1n = 1: 1!0!=11=1=n\frac{1!}{0!} = \frac{1}{1} = 1 = n. Works (using 0!=10! = 1).
  3. 3
    For n=0n = 0: 0!(−1)!\frac{0!}{(-1)!} — (−1)!(-1)! is undefined. The formula breaks at n=0n = 0.
  4. 4
    So the correct statement needs the domain restriction n≥1n \ge 1.

Answer

n!(n−1)!=n for n≥1; undefined at n=0\frac{n!}{(n-1)!} = n \text{ for } n \ge 1;\text{ undefined at } n=0
Edge cases at boundary values (like 0 and 1 for factorials) often reveal unstated domain restrictions. Always test the smallest values in the claimed domain.

About Edge Cases

Special or extreme input values — such as zero, infinity, empty sets, or boundary conditions — where formulas or reasoning may behave differently.

Learn more about Edge Cases →

More Edge Cases Examples

Example 1 easy

For the function [formula], check the edge case [formula] and describe what happens.

Example 3 easy

For [formula], identify the edge case and state the domain.

Example 4 medium

Test the edge cases [formula] and [formula] for the formula [formula].

← All Edge Cases Examples Edge Cases Concept