Suppose there is a particle moving from A to C, but that it only can move to the right or upwards a fixed distance per step. Moving the sliders you can see the different possible moves the particle has for going from A to C depending on the number of steps.

For a given maximum number of vertical/horizontal steps, how many different paths from A to C are there?

If the movement is at random, such that in every step the particle has 50%-50% chances to go to the right or upwards, what is the expected amount of steps before reaching C?