I understand the purpose for division. Multiplication is simply repeated addition and division is simply repeated subtraction. So lets say we have 4 total items. Plugging 4 into the equation we get 4(4-1)/2 = 12/2 = 6. So there are 6 possible combinations with 4 items. Applying the intuitive understanding of division as repeated subtraction, we can plot 12 on a numberline, and then since we are dividing by 2, we count backwards by 2 until we reach 0. We can do this 6 times.
That makes sense to me. My issue is with the numerator. Why must a number of items be multiplied by one number less than itself? I understand the basic reason for multiplying the number of items by itself: pairs include two items, so the 4 takes care of the possible numbers on the first item, but why does the second item only receive 3 possible combinations? My hypothesis is that it prevents overlap, but I am having difficulty interpreting the logic of this. How exactly does it overlap? Can this be visualized on a matrix? Why does 4*4/2 = 8 not give the correct number of unique pairs?