How many different integers can be formed using the digits 1, 2, 3, and 4 exactly once?

To determine how many different integers can be formed using the digits 1, 2, 3, and 4 exactly once, we can use the concept of permutations. Since we are working with four distinct digits and each digit must be used exactly once in forming the integer, we need to calculate the number of ways to arrange these four digits.

The formula for the number of permutations of n distinct objects is given by n!, which means "n factorial." For a set of four digits, n is 4. Therefore, we calculate 4! as follows:

4! = 4 × 3 × 2 × 1 = 24

This result signifies that there are 24 different ways to arrange the digits 1, 2, 3, and 4 to form unique integers. Each arrangement corresponds to a distinct integer, as the order of digits determines the integer's value.

This understanding allows us to confirm that the number of unique integers that can be formed with the digits 1, 2, 3, and 4, each used exactly once, is indeed 24.