Jeremy is fitting candies in a hexagonal box.

[thed0ctor]

Jeremy is fitting candies in a hexagonal box. He can arrange them in any configuration he wants, as long as there are 4 candies in each of the 3 orientations shown:

In how many different ways can Jeremy fit the candies?

I feel like it's a permutation combination type thing but I just can't think up a formula. I know there has to be some way other than trying to do find single combination. I've found out a few, but I feel like theres a formula or something. Like one candy can cover 2 triangles and there's 24 triangles so you can only fit 12 all together. I can't seem to find a pattern but yeah...

Any help would be appreciated, thanks!

 

[lookagain]

I counted \(\displaystyle 18\) ways.

Anchor the vertical pairs (green), and then the slanted pairs (red and blue) can have only one
orientation per arrangement of vertical pairs.

Number the vertical pairs starting from the top row and going from left to right for each row:

Row 1: 1, 2, 3

Row 2: 4, 5, 6, 7

Row 3: 8, 9, 10

There are three larger vertical diamond-shaped sets (two triangles on a side, each comprised
of a total of eight triangles) that span across the figure, and the other remaining shapes go
in only one way for each of those: {1, 4, 5, 8}, {2, 5, 6, 9}, and {3, 6, 7, 10}

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There are three vertical shapes on the leftmost center of the figure with one vertical shape on the
rightmost center corner. Similarly, for a related second way, take the mirror image of this:

{1, 4, 8, 7} and {3, 7, 10, 4}

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There are {1, 4, 8. 6} and {3, 7, 10, 5}.

There are {2, 5, 9, 7} and {2, 6, 9, 4}.

There are {4, 8, 3, 7} and {1, 4, 7, 10}.

There is {4, 2, 9, 7}.

There are {5, 9, 3, 7}, {4, 8, 3, 7}, and {4, 8, 2, 6}.

There are {2, 5, 7, 10}, {1, 4, 7, 10}, and {1, 4, 6, 9}.