Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(n>1\), then \(n !=n(n-1) !\)

Short answer

The statement, If \(n>1\), then \(n! = n(n-1)!\) is false, because the right side of the equation does not accurately replicate the factorial operation for any values of n greater than 1.

Step-by-step solution

Understanding factorial operation

The factorial of a non-negative integer n is the product of all positive integers less than or equal to n. It's denoted by n!.

Evaluate the given statement with an example

Let's take n = 4 for example. According to the statement, \(4! = 4(4-1)!\). Evaluating this would give \(24 = 4.6\), which is clearly untrue. So, the statement is false.

Explanation of the falsehood

The statement would only be true if n was equal to 1 (n = 1). This is because \(1! = 1(1-1)! = 1\), anything other than this contradicts the given claim. Thus, the statement is false for \(n > 1\).