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Can someone explain the number theory to me to derive the answer

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closed as off-topic by Hanul Jeon, Saad, Shailesh, Hurkyl, user284331 Mar 27 '18 at 1:19

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    $\begingroup$ 1) all pieces same/differnt? 2) square: four pieces with no 'gaps' between them? $\endgroup$ – Alex Mar 26 '18 at 22:18
  • $\begingroup$ Are the 16 squares fixed in advance, or do you just mean that each piece goes in a different square? $\endgroup$ – saulspatz Mar 26 '18 at 22:28
  • $\begingroup$ You should better specify what are the limitation in the arrangements $\endgroup$ – gimusi Mar 26 '18 at 22:32
  • $\begingroup$ Well, there are 64 places to put the first piece, and one placed there are 63 places to put the second piece.... $\endgroup$ – fleablood Mar 26 '18 at 22:57
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I assume you are asking about arranging $16$ chess pieces of the same color anywhere on the board, with no other pieces of the other color on the board at this time.

First we have to choose the $16$ squares that we want to choose to have a piece on it out of the total $64$ so this is $$\binom{64}{16}$$

Next we have to arrange the $16$ pieces (the 8 pawns, 2 rooks, 2 bishops, 2 knights, the queen, and king). This is $$\frac{16!}{8! 2! 2! 2!}$$

Multiplying these two quantities gives a grand total number of ways to be $31688202068279540784000$ or roughly $31.7 \text{ sextillion}$

Just for size reference, one sextillion is roughly equal to $3$ times the amount of gallons of water on Earth (Source)


Edit: I forgot to mention that in this case the orientation of the board is fixed as well. If we want to say that two particular arrangements that only differ by rotating the board $90°$, $180°$, or $270°$ are the same arrangement, then we need to divide our final answer by $4$.

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