We are trying to arrange the numbers. There are 10 numbers to arrange, zero to nine. There are only four spaces, or slots, for each number. In the first space, place a 10 because there are 10 numbers; in the second space, place a nine; in the third space, place an eight; and in the fourth space, place a seven. Multiply 10 times nine times eight times seven.
The total number of all such possible passcodes is 10∗9∗8∗7=5040.
Following is a demonstration of the solution. The difference between the demonstration and four-digit PIN guessing is that we have 10 numbers to choose from, not four letters, as in the demonstration. Note that 0! = 1.
In every math session I have tutored, we did not do any New Math. One student showed me some New Math, and we solved the problem the old way. Ha!