<PRE><B>Question 1133068</B>
&lt;br&gt;
Determine the total number of paths from S to F, then subtract the number of paths from S to F that do NOT cross the main diagonal.&lt;br&gt;
(1) To get from S to F moving only right (&quot;R&quot;) or down (&quot;D&quot;), you need to have the moves&lt;br&gt;
RRRRRRDDDDDD&lt;br&gt;
in some order.  The number of different ways of arranging those letters, and therefore the total number of paths from S to F, is&lt;br&gt;
{{{12!/((6!)(6!))}}} =  12C6 = 924.&lt;br&gt;
The paths that do NOT cross the main diagonal are of two types: those that always stay below or on the main diagonal, and those that always stay above or on the main diagonal.&lt;br&gt;
There is a symmetry between those two sets of paths; the numbers of paths are the same for both types.  So count the number of paths of one of the types; the total number of paths that do not cross the main diagonal is double that number.&lt;br&gt;
To count the number of paths that always stay on or below the main diagonal, draw a picture of the grid (or at least that part of the grid) and label each intersection on the grid with the number of ways to get there:&lt;br&gt;&lt;pre&gt;

   +---X
   |   |
   1---1---X
   |   |   |
   1---2---2---X
   |   |   |   |
   1---3---5---5---X
   |   |   |   |   |
   1---4---9--14--14---X
   |   |   |   |   |   |
   1---5--14--28--42--42---X
   |   |   |   |   |   |   |
   1---6--20--48--90--132-132&lt;br&gt;&lt;/pre&gt;
-------------------------------------------------------&lt;br&gt;
Here is a quick description of how those numbers are obtained, in case you don&#39;t see it....  Look at the row of numbers 1 - 3 - 5 - 5.&lt;br&gt;
(1) The 1 can only be reached by moving down; there is only 1 way to get the the intersection above, so there is only 1 way  to get here.
(2) The 3 can be reached by moving either right or down.  There is 1 way to get to the intersection to the left, and there are 2 ways to get to the intersection above; so there are 3 ways to get here.
(3) The first 5 can also be reached by moving either right or down.  There are 3 ways to get the the intersection to the left, and there are 2 ways to get to the intersection above; so there are 5 ways to get here.
(4)The second 5 is on the main diagonal, so you can&#39;t get here by moving down; you can only get here by moving right.  There are 5 ways to get to the intersection to the left, so there are 5 ways to get here.&lt;br&gt;
----------------------------------------------------&lt;br&gt;
We now know that there are 132 paths from S to F that stay on or below the main diagonal; by symmetry there are 132 paths from S to F that stay on or above the main diagonal.&lt;br&gt;
So we have a total of 924 paths from S to F with no restrictions; 132+132=264 of them stay either on or below the main diagonal, or on or above the main diagonal.&lt;br&gt;
So we have our answer:&lt;br&gt;
The number of paths from S to F that cross the main diagonal at least once is 924-264 = 660.</PRE>