TITLE: What is the number of positions of a 15 puzzle (4x4 board)
QUESTION [7 upvotes]: I wonder if someone can help me get a number for possible positions for this childs game of 15 sliding pieces in a 4x4 board.
The normal game has numbers 1-15 that need to be sorted. The board I want to calculate has only three kind of pieces: 6 red, 6 green, 3 black, and of course the empty slot you can move through.
I referred to this post and came up with this solution $\binom{16}{6}\binom{10}{6}\binom{4}{3}=6.726.720$. Does that look correct, or did I miss something?

REPLY [5 votes]: You are correct, and you should not divide by $2$ as in the ordinary 15 puzzle because it is possible to swap two pieces of different colours without changing the appearance of the rest of the puzzle – simply swap in addition two pieces of the same colour, effectively doing nothing at all with this second swap.