Consider an ant that is walking on a Cartesian grid, starting at (0,0) and ending at (10, 15). The ant always chooses to walk exactly one unit either up or to the right (towards his destination) whenever he arrives at a Lattice point. (A Lattice point is a point with integer coordinates.) Thus, from (0,0) he either walks to (1, 0) or (0, 1). How many different paths can he take on his walk?
The answer to this question is C(25,10) = 3,268,760
I understand that my n = 25 because there are a total of 25 moves possible but why would r = 10? I'm just a bit lost on how to choose the values for these variables.
You can think of a sequence of $25$ steps where each one is either "up" or "right, with exactly $10$ instances of "right" and $15$ instances of "up". So, to count paths, you just need to count these sequences.
But, to count such sequences, you just need to note that once you've determined the $10$ positions which contain "right", the rest of the remaining positions are ALL going to be "up". So, $\binom{25}{10}$ counts the paths you are looking for.