Sample question.
6. In how many ways can one travel from (0,0) to (8,11) going only East or North, and while passing through (4,7)?
Solution
1) There are (11 choose 4) ways to go from (0,0) to (4,7). The explanation: you have to do 11 moves, 4 of them should be east (alternatively 7 of them should be north).
2) There are (((8-4)+(11-7)) choose (8-4))=(8 choose 4) ways to go from (4,7) to (8,11). The explanation: you have to do 8 moves, 4 of them should be east (alternatively 4 of them should be north).
3) By product rule (you can pair up each way from 1) with a way from 2) ) total number of ways to go from (0,0) to (8,11) through (4,7) is (11 choose 4) * (8 choose 4) = 23100
