Question

Referring back to problem 8 (second picture posted below), how many paths are there from (1,0) to (5,7) if there are no restrictions on the move D?

Answer 1

To reach the point (5, 7) form (1, 0) one has to traverse 5 - 1 = 4 steps to the right (i.e. 4 R moves) one has to traverse 7 - 0 = 7 steps to the up (i.e., 7 moves) Already move D has made k times Hence there are remaining 4-k R steps & 7 - k, u steps to reach (5, 7) form (1, 0) (a) 4 - k R steps (b) 7 - k steps (c) Number of paths from (1, 0) ti (5, 7) = number of permutations of 11 steps (i.e., 4R + 7 U) of which 4 - k step are of the same type (namely R) & 7 - k are of the same type (namely U) = 11!/(4 - k)! (7 - k)!