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June 12th, 2014, 10:11 PM   #1
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total cuboids possible based on minimum and maximum dimensions

Lets assume that all the dimensions are integers. Lets also assume that the minimum dimension of any side is 1 and maximum is 17.

How many cuboids are possible excluding perfect cubes and which ones are they?

I think it is (17*17*17) - 17 or 17*17*16 which is another way of saying it for the total possibilities - cubes.

Why?

Well for any 2 dimensions they can be the same so at least 1 has to be different.

There are also 17 integer cubes possible from 1x1x1 to 17x17x17 so because I am wanting cuboids that aren't cubes I subtract that from the total possibilities.

Last edited by caters; June 12th, 2014 at 10:16 PM.
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June 14th, 2014, 01:13 PM   #2
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I going to assume I'm interpreting your question correctly.

You're also going to have to consider repeats. For example, a 2x3x4 cuboid is the same as a 3x2x4 cuboid and a 4x3x2 cuboid. There are 6 duplicates for some cuboids (specifically those with all sides distinct), and only 3 duplicates for others (those with two sides the same).

Luckily, there's a formula for combinations with repeats. If you're choosing $r$ objects out of a total of $n$ things, and order doesn't matter but repetition is allowed, then the total number of ways to do this is

$\displaystyle \binom{n+r-1}{r}$

In your case, $n=17$ and $r=3$, so the total number of cuboids that fit your criteria is

$\displaystyle \binom{17+3-1}{3}-17=\binom{19}{3}-17=952$

Hope this helped.
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