Question 946288
<pre>
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.

{{{drawing(200,150,-1,7,-1,5,
locate(-.3,4.5,A),locate(6,0,B),

line(0,0,6,0),line(0,1,6,1),line(0,2,6,2),line(0,3,6,3),line(0,4,6,4),
line(0,0,0,4),line(1,0,1,4),line(2,0,2,4),line(3,0,3,4),line(4,0,4,4),line(5,0,5,4),line(6,0,6,4) )}}}

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</pre>