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weirdojuwin
20.07.2019 •
Mathematics
From (0,0,0) to (3,3,3) in 3d space how many paths are there if we move only right, forward or up?
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Ответ:
Consider the image attached to the answer. The number in each square signifies the ways to reach that square.
At our starting square
Let us look at the square above
However, it gets more interesting for the remaining squares. At
There is a quicker way to find the possibilities. Since the only way to reach a certain square is either from a square below or to the left, the possibilities of reaching a square are equal to the sum of the possibilities of reaching a square below it and the possibilities of reaching a square to the left of it.
For example, look at square
Thus, for this problem, the answer is
Now, let us look at your problem, which is the 3d version of this problem. You can apply the same ideas, that the possibilities of reaching each square are equal to the sum of the possibilities of reaching the squares below, to the left, and backward.
It may take some time to get used to 3d (draw four 4-by-4 grids), but you should get the answer of
Advanced
If you notice carefully, for the 2d problem, the grid forms Pascal's triangle. The
On our grid, the square with coordinates
Applying the formula for Pascal's triangle, we have
Substitute
You can look up Pascal's pyramid to solve the 3d version of the problem. The formula you should get is
Substitute
Ответ:
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