How many different paths can the robot take?

Question

There is a 10 x 10 grid. I'll indicate the intersection of grid lines, considering the bottom-left one to be (0, 0) and the top-right to be (10, 10). A robot takes either a vertical line up or horixontal line right each second to reach the top-right in 20 moves. However the point (4, 5) is forbidden. The robot gets the electric chair it it gets to this point. ;)

How many different paths exist if the robot has to come through without getting the chair?

Please come up with a valid explanation for your answer.

Answer 1

Firstly, the number of ways of going from (0,0) to (a,b) is (a+b)!/a!b!. This is because you have to make a total of a+b moves, a of which are up and b of which are across. You should be able to convince yourself that this is true.

Now the problem is easy. You want the number of paths which reach (10,10) but which don't pass through (4,5), so this is the same as the number of all the paths which reach (10,10) minus the product of the number of paths to (4,5) times the number of paths from (4,5) to (10,10). All three of these numbers can be worked out using the above formula.

Answer 2

to have exactly 20 steps, the only option is 10 steps Up and 10 steps Right, the total number of options to arrange exactly 10 U + 10 R in any order of appearance is?
out of which every option where the first 9 steps are exactly 4 U and 5 R are excluded.