Counting ways on the coordinate system

Question

What is the number of ways in which you can reach a point (2,2) from the origin, taking unit steps at a time , in not more than 7 steps?

Answer 1

You need to have 2 right, and 2 up, and remaining 3 moves have to cancel to 0, so you have 2 right, 2 up, and then either 1 up 1 down, or 1 right 1 left, or nothing at all.

For 6 moves, by symmetry, the correct number is twice the case where you use 1 up and 1 down, and now we're choosing how to order 3 up, 1 down, and 2 right, which should be the multinomial coefficient $$\frac{6!}{3!1!2!}$$So you want twice that, which is 120.

For 4 moves, it's similarly $\frac{4!}{2!2!} = 6$, so you get a total of 126.