Cannot find expression for number of paths

1. Random walks. A famous carnival game is a machine that drops a large number of marbles at the top of a pyramid of nails hammered partly into a board like this: marbles dropped here bins The marbles fall down and at each step have a 50% chance of going right or left. At the bottom they fall into a set of bins as shown. If there are N rows of nails total, there are N + 1 bins. a) Write down an expression for the number of paths g(N; l; 7') that a marble can take that make a total of I steps to the left and r steps to the right. Then eliminate t and r in favor of the distance x = r l traveled to the right to get the same number in terms of N and a: only. Note: the distance a: is measured horizontally from where the marbles start, i.e. from a line down the middle of the picture above. Also, :1: changes by increments of 2. For example, if 'r = 5 and l = 4, then .7; = 'r l = 2. If we ip one step from left. to right, we have 1" = 7 and l = 3, which implies x = 4. As long as N = T +1 is even, x has to be even, and there's no way to get :1? = 3. Finally, you might nd it helpful to write out explicitly for yourself how r, E, and a: are related to the variables in our standard spin system (but this is not required)