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Chapter 11: Problem 73

Express \({ }_{n} P_{n-2}\) without using factorials.

Short Answer

Expert verified
The simplified form of \({ }_{n} P_{n-2}\) without using factorials is \(n \times (n-1)\).

Step by step solution

01

Definition of Factorial

Factorial is a mathematical function symbolized by '!'. It is a product of a non-negative integer 'n' with all the non-negative integers below it. For example, \( 4! = 4 \times 3 \times 2 \times 1 \). The key here is to remember that \((n-2)! = (n-2) \times (n-3) \times ... \times 1\) and \(n! = n \times (n-1) \times (n-2) \times (n-3) \times ... \times 1\). Notice that the term \((n-2) \times (n-3) \times ... \times 1\) exists in both, hence they cancel each other.
02

Simplifying

We then simplify the permutation \({ }_{n} P_{n-2}\) = \(\frac{n!}{(n-2)!}\) by noticing that the factorial terms can cancel out. This leaves us with \(n \times (n-1)\)
03

Final Simplified Form Without Factorials

The simplified permutation is \(n \times (n-1)\) which does not contain any factorials, as we were tasked to achieve.

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