Question

Skipper follows a path along the grid shown, always moving either up or

to the right. How many paths from A to C do not pass through B?

Answer 1

We can use complementary counting. First we can count ALL the ways we can get from A to C. Then we can count the ways we can get from A to C, where we DO pass through B. Then we do the total minus the number of ways we DO pass through B.

Number of TOTAL ways from A to C.
First we can imagine fake, dotted lines around the top left and bottom right "corners" so we can make a complete square.
Once we have this "complete" square with the dotted line corners, we can easily find the number of paths from A to C. It is just
`(8!)/(4!*4!)` = 70. If you don't know how this was derived, just comment.
HOWEVER, of the 70 paths we just counted, 2 of them are not possible using the original picture. Remember those dotted corners we drew? We cannot create paths using those corners; they were just there so we could use a formula. So, we actually have 70-2 = 68 total paths.

Now we find the number of paths from A to C that DO pass through B.
Well, first we have to find the number of paths from A to B and then from B to C. This will guarantee we move through B. From A to B, there is
`(4!)/(2!*2!) =6` ways. From B to C, there is also
`(4!)/(2!*2!) =6` ways. 6*6 = 36

68-36 = 32 FINAL ANSWER