# Counting ways on the coordinate system

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?

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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.