Factorial Math Example 4

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Example 4

medium
Solve for nn: n!(n−2)!=30\frac{n!}{(n-2)!} = 30.

Solution

  1. 1
    Expand the factorial quotient: n!(n−2)!=n(n−1)\frac{n!}{(n-2)!} = n(n-1).
  2. 2
    Set up the equation: n(n−1)=30⇒n2−n−30=0n(n-1) = 30 \Rightarrow n^2 - n - 30 = 0.
  3. 3
    Factor: (n−6)(n+5)=0(n-6)(n+5) = 0, so n=6n = 6 or n=−5n = -5.
  4. 4
    Since factorials use nonnegative integers here, n=6n = 6.

Answer

n=6n = 6
Factorial quotients often simplify by canceling common factors. After simplification, the problem becomes a standard quadratic equation.

About Factorial

The factorial of a non-negative integer nn, written n!n!, is the product of all positive integers from 1 to nn: n!=n⋅(n−1)⋯2⋅1n! = n \cdot (n-1) \cdots 2 \cdot 1.

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More Factorial Examples

Example 1 easy

Compute [formula].

Example 2 medium

Simplify [formula].

Example 3 easy

Evaluate [formula].

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