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I have three kinds of items (infinite supply): $a$, $b$ and $c$. I want to choose freely from the infinite large collection of items and put them in $n$ number of boxes. However: Only one item is allowed in each box and I am not allowed to put item a in the first box. Neither am I allowed to let neither $a$:s nor $c$:s be in adjacent boxes. What are all the possible combinations I can get and how many permutations are there for each combination if there are $n$ number of boxes?

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  • $\begingroup$ What did you try? Do you have any thoughts. If yes then please share because asking a question without showing your work might attract downvote. $\endgroup$ Commented Jan 30, 2018 at 3:03
  • $\begingroup$ My comment was too long to be posted here. ... $\endgroup$ Commented Feb 6, 2018 at 0:00

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Well... I made a series of matrices and tried to figure out the patterns. Since b is allowed to be placed anywhere, let us let the exponent of b be equal to the total number of variables minus the sum of exponents of the other two variables. I found, that if one of the variables a and c are non-present the coeffcient can be derived from Pascal's triangel. All 1:s have coefficents 1,2,3,4,5... all 2:s 1,3,6,10,15 ... all 3:s 1,4,10,20... etc. If a and c are 1, then the coeffeficient is a perfect square. If there are 2 c:s and 1 a they seem to be 1*2, 3*3, 6*4, 10*5... and 1*3, 3*4, 6*5, 10*6 ... for 3 a:s. If there are 2 c:s and 1 a they are 1 , 5, 15 ... As for a^2c^2, I beleive they are 1, 7, 28 ...

So ... There are clearly patterns here, but it is hard to work it out without running a simulation. This is actually part of a larger probability problem I am working with. I really should learn programming! The online calculators are very good, but you can't make algorithms for solvning problems and making many simultanious calculations. Online calculators also collaps when the expressions you put in get too massive.

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