Combination of pieces on Chess board

Puzzle # 10

How many ways are there to arrange the sixteen black or white pieces of a standard international chess set on the first two rows of the board?

1. Each square can hold only one chess piece
2. All eight Pawns are identical
3. Similarly each rook, knight and bishop are identical to its pair.

What could be the possible combinations to arrange the chess pieces on chess board?


Solution
Total available squares to arrange pieces on chess board= 16

       6,48,64,800 WAYS !!

There are total 16 pieces which can be arranged on 16 places ¹⁶P₁₆ = 16!

(16! = 16 * 15 * 14 * 13 * 12 * ..... * 3 * 2 * 1)but, there are some duplicate combinations because of identical pieces.

There are 8 identical pawn, which can be arranged in ⁸P₈     = 8!

There are 2 identical Bishops, which can be arranged in ²P₂ = 2!

There are 2 identical Rooks, which can be arranged in ²P₂   = 2!

There are 2 identical Knights, which can be arranged in ²P₂  = 2!

Hence, the require answer is = (16!) / (8! * 2! * 2! * 2!)
                                            = 6,48,64,800

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