How many paths are there from a to b ​, if every step must be up or to the right?

We  have  all  the possible  arrangements  of  this set   

{right, right, right, right, right, up, up, up up }

We can either  choose  any 5 of the 9 places to put  the "rights"  or  any 4 of the 9 places to place the "ups"

So

C(9,5)  =   C(9,4)   =    126 paths

the answer is 126 because in the end you are going to have to move 9 steps in all. So, 9 choose 5 equals 126.