How many different paths are there from the point (0, 0, 0) in three dimensional space to the point (7, 3, 4) if you are only allowed to move in steps of one unit in either the positive x direction, the positive y direction, or the positive z direction. (That is no backtracking is allowed and you can only move in a single direction on a given step)

Using the fundamental counting principle, it is found that there are 84 different paths.

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Fundamental counting principle:

States that if there are p ways to do a thing, and q ways to do another thing, and these two things are independent, there are p*q ways to do both things.

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The movements are independent, which means that the fundamental counting principle is used.

There are:

Thus, the number of different paths is:

A similar problem is given at

3: 2

step-by-step explanation:

shorter piece is 2: 3, so just reverse it, and you get 3: 2