Sammy the spider

The general solution of this type of problem for the number of ways $N(m,n) = \dbinom {m+n}m = \dbinom {m+n}n$ , where $m$ and $n$ are integral horizontal distance and integral vertical distance between the starting point and the ending point respectively. Therefore, for $m=4$ and $n=3$ , we have $N(4,3)=\dbinom {4+3}3 = \dbinom 73 = \boxed{35}$ .

We can also solve the problem using a diagram shown above by assigning at each node of the grid the number of ways to the destination. That numbers for nodes on the top horizontal line are all 1's $N(m,0) = \dbinom m0 = 1$ . Similar those of the vertical line on the right $N(0,n) = 1$ . For the rest of the nodes, the number is the sum of the number above and the number on the right or $N(m,n) = N(m, n-1)+N(m-1,n)$ . We find that all $N(m,n) = \dbinom {m+n}n$ .