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

c.   x=2.4cm

step-by-step explanation:

similar triangles are triangles which have the same shape but not the same size. this means their angle measures are equal but their side lengths are not instead they are proportional. to solve, we will set up a proportion and solve.

a proportion is two equal ratios set equal to each other. we form the ratios by finding corresponding sides (sides which match to each other on both triangles by their position). out ratios will be one side of the small triangle over the corresponding side on the big triangle as shown below:

.

to solve the proportion, we'll cross multiply by multiplying numerator with denominator across the equal sign.

step-by-step explanation:

because it just is