To simplify the expression [tex]\(\frac{(n-4)!}{3!(n-2)!}\)[/tex] and evaluate it for [tex]\(n = 6\)[/tex], follow these steps:
1. Expression Analysis:
- The given expression is [tex]\(\frac{(n-4)!}{3!(n-2)!}\)[/tex].
- Recall that the factorial function [tex]\(k!\)[/tex] represents the product of all positive integers up to [tex]\(k\)[/tex].
2. Rewrite the Factorials:
- Notice that [tex]\((n-2)!\)[/tex] can be expanded as [tex]\((n-2)! = (n-2)(n-3)(n-4)!\)[/tex].
- Substitute this expanded form into the original expression:
[tex]\[
\frac{(n-4)!}{3!(n-2)!} = \frac{(n-4)!}{3! \cdot (n-2)(n-3)(n-4)!}
\][/tex]
3. Cancel Common Terms:
- The [tex]\((n-4)!\)[/tex] in the numerator and denominator cancel each other:
[tex]\[
\frac{(n-4)!}{3! \cdot (n-2)(n-3)(n-4)!} = \frac{1}{3! \cdot (n-2)(n-3)}
\][/tex]
4. Simplify the Constant Factorial:
- Evaluate [tex]\(3!\)[/tex]:
[tex]\[
3! = 3 \cdot 2 \cdot 1 = 6
\][/tex]
- Substitute [tex]\(3!\)[/tex] with 6 in the expression:
[tex]\[
\frac{1}{6 \cdot (n-2)(n-3)}
\][/tex]
5. Thus, the simplified form of the expression is:
[tex]\[
\frac{1}{6(n-2)(n-3)}
\][/tex]
6. Evaluate the Expression for [tex]\(n = 6\)[/tex]:
- Substitute [tex]\(n = 6\)[/tex] into the simplified expression:
[tex]\[
\frac{1}{6(6-2)(6-3)} = \frac{1}{6 \cdot 4 \cdot 3} = \frac{1}{72}
\][/tex]
7. Conclusion:
- The simplified form of the given expression is:
[tex]\[
\frac{1}{6(n-3)(n-2)}
\][/tex]
- When [tex]\(n = 6\)[/tex], the value of the expression is:
[tex]\[
\frac{1}{72}
\][/tex]
