Answer to Question #38563 in Quantitative Methods for Sheila Koomson

Question

Suppose you have a wire mesh which is N by M units long, made up of unit square with wire at the edges. (So there are N+1 parallel wires all M long and, perpendicular to these, M+1 all N long).An ant starts off at the bottom left corner of this grid (co-ordinates (0,0) and crawls on the wires the shortest possible distance to reach the top-right corner (N,M). How long is the shortest route. How many different shortest routes are there? (Namely, find a formula in terms of N and M) You might want to try this for small values of N and M and see if you can work out how the number for (N,M) relates to those for (N,M-1) and (N-1,M) If you build up a table of these numbers you might recognise them from elsewhere in mathematics so that might help you find the formula, but you also try to explain the connection.