# How Many Paths?

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.

1. 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:
2. 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:
3. 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.
4. 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.
5. 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)?
6. 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)?
7. 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)?
8. 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.