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?

The formula n(n−1)/2 for the number of pairs you can form from an n

element set has many derivations, even many on this site.

One is to imagine a room with n

people, each of whom shakes hands with everyone else. If you focus on just one person you see that she participates in n−1 handshakes. Since there are n people, that would lead to n(n−1)

handshakes.

Note here that we are thinking about the multiplication xy

as the total for x groups of y

things, not as repeated addition. Computationally they are the same, but psychologically they are different.

To finish the problem, we should realize that the total of n(n−1)

counts every handshake twice, once for each of the two people involved. So the number of handshakes is n(n−1)/2

.

Note (continued). We are thinking about x/y

as the number of groups of size y it takes to get to a total of x

, not as repeated subtraction. Computationally they are the same, but psychologically they are different.

Among the other ways to think about this problem (and one which kids find first). Imagine the people entering the room one at a time and introducing themselves. The first person has nothing to do. The second person shakes one hand. The third person shakes with the two already there, and so on. The last person has n−1

hands to shake. So the total number of handshakes is

0+1+⋯+(n−1).

There are many tricks for evaluating that sum to discover that it’s n(n−1)/2.