A classic problem built off the Cartesian plane and moving from one point to another. Rather than asking students explicitly for the formula for paths from (0, 0) to (m, n), I think it is more important to ask about an m and n sufficiently large to demand students come up with a formula/strategy because counting, especially by hand, might take longer than the life of the universe.

- Given that you are restricted to the directions up and right, how many possible paths are there from (0, 0) to (2, 1). See below for a visual:
- Given that you are restricted to the directions up and right, how many possible paths are there from (0, 0) to (2, 1). See below for a visual:
- Given that you are restricted to the directions up and right, how many possible paths are there from (0, 0) to (20, 20). Go to Project Euler: Lattice Paths and type in your solution to check your answer.
- Your answer to problem 3 certainly did not come from counting all possibilities, so explain your strategy, including any incorrect lines of reasoning you pursued.
- Suppose you are restricted to moves in the positive x, y, and z directions. How many paths are there from (0,0,0) to (1, 1, 1)?
- Suppose you are restricted to moves in the positive x, y, and z directions. How many paths are there from (0,0,0) to (1, 2, 2)?
- Suppose you are restricted to moves in the positive x, y, and z directions. How many paths are there from (0,0,0) to (10, 10, 10)?
- Did your strategy from problem 4 work? If not, were you able to adapt it or did you have to start from scratch? Similarly, explain your strategy including any incorrect lines of reasoning you pursued.

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