Question

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

Answer 1

If we let numbers repeat `= 256`. If we don't let numbers repeat `=24`. If we're talking strictly about combinations (vs permutations) `= 1`.

Explanation:

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.