If we have the numbers 1, 2, 3, and 4, how many combinations of 4 digit numbers can we make?

If we are looking at the number of numbers we can create using the numbers 1, 2, 3, and 4, we can calculate that the following way:

• for each digit (thousands, hundreds, tens, ones), we have 4 choices of numbers. And so we can create #4xx4xx4xx4=4^4=256# numbers

If we are looking at the number of numbers we can create using the numbers 1, 2, 3, and 4 but without repeating numbers, we can calculate that the following way:

• in the thousands place, we have 4 choices (1, 2, 3, 4). In the hundreds place, we'll then have 3 choices (1, 2, 3, 4, less the one taken for the thousands). And then for the hundreds we have 2 choices, and the ones have the remaining choice. That gives us #4xx3xx2xx1=4! =24# numbers

If we're talking strictly about combinations (vs permutations), where the order of picking the elements doesn't matter (much like in a poker hand where the 1, 2, 3, 4 of spades is the same as the 4, 3, 2, 1 of spades), we can calculate that the following way:

• we see that we have 4 possible numbers and we're picking all 4, so no matter what we do we'll always pick the same 4 numbers (1, 2, 3, 4) which means there is one 1 combination.