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1. There are 2 paths from point (0,0) to (2,2), under the line of y=x. As we have said last time in math circle we need 2R (Rights) and 2U (Ups) for us to get to the point (2,2).

The possible paths were 6.(Including lines on y=x)

The paths are:

1.RRUU

2.RURU

3.URUR

4.URRU

5.RUUR

6.UURR

In this problem it doesn’t let us go one the line y=x. So that means our first move cannot be U or RUU. That eliminates #s 3,4,5,6. That leaves us with 2 paths.

You can also do it with the Catalan number formula:(2n choose n)/n+1.In this case n=2.

(4 choose 2)/3.

4 choose 2 is 6. And, 6/3=2. Leaves us again with 2 possible paths.

For the possible paths from (0,0) to (3,3) below the line of y=x.

We can use the Catalan number formula to be more efficient.

(2n choose n)/n+1

In this case n=3.

(6 choose 3)= 20

20/(3+1)

20/4

=5

Leaves us with 5 possible paths.