Help!

The top right corner is Mack's goal. Bottom left is where he starts and the red spot is the spider. Blue numbers are the number of ways Mack could get to the point.

He starts at (0,0) and there is only one way to get there. There is also only one way for him to get to all the points on the bottom and left edges, because he can only move up and right.

To get to (1,1), there are 2 ways to get there because he could come from (0,1) or (1,0).

To get to (2,1) he could come from (2,0) or (1,1). The number of ways he could get to (2,1) is the sum of the ways he could get there from the point below or the point to the left so 2+1=3 ways to get to (2,1).

If you look at the drawing, all the blue numbers are the sum of the blue numbers 1 below and 1 to the left. You might notice that this looks like Pascal's triangle! (Because it pretty much is).

Continuing to sum the numbers in this manner, the number of ways to get to (5,7) is \(\boxed{426}\)

There are other ways to solve this problem using combinations, and if this is a homework problem that is probabally the way you know how to solve it. I like this way better .

I think the way to solve this is to count all the paths that go from (0,0) to (5, 7)   and then eliminate those that pass through (2, 3)

The paths that go from (0, 0) to (5, 7)  are

(R , R, R, R, R, U , U, U, U , U, U , U)   = C(12,5) = C(12,7)  = 792

The paths that go from (0,0) to ( 2,3)  =

(R, R, U, U, U)  =  C(5,2) = C(5,3)  = 10

And the paths that go from (2,3) To (5, 7)   =

(R, R, R, U, U , U , U)  = C(7,3) = C(7, 4) = 35

So...the total paths that go from (0,0) through (2, 3) and then from (2, 3) to (5,7) = 10 * 35  = 350

So.....the total paths from (0,0) to (5, 7)  that avoid going through (2, 3)  =  

792 - 350  =

442

Mack the bug starts at (0,0) at noon and each minute moves one unit right or one unit up.

He is trying to get to the point (5,7). However, at (2,3) there is a spider that will eat him if he goes through that point.

In how many ways can Mack reach (5,7)?