If  is the number of different paths of length-12 which leads from  to  in the grid which do not pass through , then the value of , where [ .] represents the greatest integer function, is

 

(5)

Here the path which leads from A to B is of length-12

Now without considering the constrain of passing through the point M, number of ways in which we can reach B from A is equal to number of ways we can select 6 steps from left to right and 6 from bottom to top which is equal to

Now we can reach from A to M is 6 steps in  ways and can reach from M to B in ways

Hence we canreach from A to B through M in  ways

Hence required number of ways