I think what you want is how many ways you can get from A to B, on the 6x4 grid
below, if you can only go right or down along the lines of the grid.   I don't
think the numbers in Pascal's triangle have anything to do with it.  Correct me
in the thank-you note if I am interpreting that wrong.



The number of rightward moves must add up to 6.
The number of downward moves must add up to 4.

The number of partitions of a number n is the number of ordered
aequences of positive integers whose sum is n. 

The number of partitions of n is given by the formula C(2n-1,n-1)

The number of possible rightward moves is the number of partitions of 6,
which is C(2*6-1,6-1) = C(11,5) = 462
The number of possible downward moves is C(2*4-1,4-1) = C(7,3) = 35

Answer 462x35 = 16170 ways.   

Edwin