n!≤m≤(n+1)!

To find the range of values that m can take for a given value of n, we need to consider the factorial function.

n! represents the product of all positive integers up to and including n. For example, 4! = 4 × 3 × 2 × 1 = 24.

(n-1)! represents the product of all positive integers up to and including n-1. For example, 3! = 3 × 2 × 1 = 6.

Using these definitions, we can rewrite the inequality:

n! ≤ m ≤ (n-1)!

as

(n × (n-1) × ... × 2 × 1) ≤ m ≤ ((n-1) × (n-2) × ... × 2 × 1).

This means that m must be greater than or equal to the product of all positive integers up to and including n, and less than or equal to the product of all positive integers up to and including n-1.

For example, if n = 4, then n! = 4 × 3 × 2 × 1 = 24 and (n-1)! = 3 × 2 × 1 = 6. Therefore, the range of values that m can take is:

24 ≤ m ≤ 6

This inequality is impossible, so there are no values of m that satisfy the inequality for n = 4.

In general, for any n ≥ 2, the range of values that m can take is:

n! ≤ m ≤ (n-1)!

This inequality holds because the product of all positive integers up to and including n is greater than the product of all positive integers up to and including n-1. Therefore, the range of values that m can take becomes smaller as n increases.