So it comes to pass once that in Academic Team, there was a question about how many high fives can be given in a room with X people if no two people give a high five twice. I seem to remember having these in earlier math. Unfortunately, instincts guide me to use factorial of X, which is most definitely not right for this situation.

Rather than multiplying the numbers below X, they are added for this. It's also like the number of cannonballs you have if a triangle (not a pyramid mind you, that gets harder) has X cannonballs at the base and narrows to one. Like so:

___o

_'_oo

__ooo

_'oooo

I studied this over the weekend, accidentally, while working with acceleration. If you want to know how far an object will fall (with an acceleration of X ft/s²) in Y seconds, you use that triagular thing of Y (and multiply by X). This is because in the first second, the object falls X feet. And the second second, it will fall 2X feet...and so on. With some work, I found that the equation for finding the triangular of a number is T_N = (not my image, but I found the same equation).

So today during practice, we had a problem about how many ways you can arrange X people in a line. And I used the triangular for once. This time, when I should have used the factorial. >.<