Can someone please explain why 1+2+3+...+n = (n+1)C2? Of course I know that they both equal n(n+1)/2. What I am really asking is why is choosing 2 objects from (n+1) objects the same as adding up the 1st n positive integers?
Being able to prove that two are equal and seeing why the two are equal are not always the same!
There isn't always a direct answer to a "why" question. But here, at least we can see that both expressions answer the same question.
That question is, given n+1 items, how many pairs are there? As it is commonly asked, if there are n+1 people in a room, and everyone shakes hands with everyone else exactly once, how many handshakes are done? Or, if you put n+1 dots equally spaced around a circle and connect each one to all the others, how many lines do you draw?
Obviously
(n+1)C
2 answers the question.
But now let's count the pairs. The first person pairs with the n other people. The second person pairs with the first (that pair is already counted) and also with the n-1 people remaining. Continue like that, and the next to last person pairs with the last person; the last person's shakes are already counted.
So the total is n + (n-1) + (n-2) + ... + 2 + 1, which is your sum.
This could be made into a nice colored animation, and probably someone has done it. I didn't invent this!