### Brain Teasers

# Mad Ade's Chess Problem Again

Mad Ade was standing outside waiting as usual for the Sweaty Chef Kebab Shop to open and was staring into the shop window of the store next door. He noticed a chess set on display.

He wondered how many ways are there of arranging the sixteen black or white pieces of the chess set on the first two rows of the board?

Obviously, Mad Ade did not bother working it out as the Kebab shop opened almost immediately as he thought about it.

What would have been the answer if Mad Ade wasn't so greedy and had worked it out?

Given that each pawn is identical and each rook, knight and bishop is identical to its pair.

He wondered how many ways are there of arranging the sixteen black or white pieces of the chess set on the first two rows of the board?

Obviously, Mad Ade did not bother working it out as the Kebab shop opened almost immediately as he thought about it.

What would have been the answer if Mad Ade wasn't so greedy and had worked it out?

Given that each pawn is identical and each rook, knight and bishop is identical to its pair.

### Answer

For 16 pieces that are all different the answer is 16! (!=factorial ie. 16x15x14...x1)But, we have duplicate combinations because there are identical pieces being used. The number of duplicate combinations is =

2 (for Rooks) x 2 (for Knights) x 2(for Bishops) x 8! (for Pawns).

This gives 8x8! = 322560.

Dividing 16! by 322560 gives us the number of unique combinations for a normal chess set = 64864800 different ways.

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