(n+!)(n−!)

We develop expressions for the probability distribution of end-end distances of a polymer chain. We start from a simple random walk in one dimension, and generalize the result to three dimensions. In one dimension, the number of ways of arriving a distance x from the origin after N steps of unit size (n+ in the positive direction and n − in the negative) is given by a combinatorial expression. W (N, x) = N! (n+!)(n−!) (1) where x = n+ − n−, N = n+ + n − and so n+ = (N + x)/2, and n − = (N − x)/2. The probability of this occurrence is just the number of ways of realizing the occurrence divided by the total number of possible trajectories. p(N, x) =