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Yash Agarwal

1. There are two paths from (0,0) to (2,2) if we have to stay behind the line y = x. Here are the two possible combinations:

RRUU

RURU

To prove this further, we can use the Catalan Number formula. (2n choose n)/n+1. Here, n=2. 4 choose 2 = 6. 6/3 = **2**.

There are five paths from (0,0) to (3,3). Here are the five possible combinations:

RRRUUU

RURURU

RRUURU

RRURUU

RURRUU

To prove this further, we can use the Catalan Number formula. (2n choose n)/n+1. Here, n=3. 6 choose 3 = 20. 20/4 = 5.

As you can see, you cannot just do n*2 choose n, since all points have to be under the line y = x. So, the answers are **2 and 5 **.

** This response was written by Yash Agarwal. **