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

Monday, March 21, 2011

Question:

How many ways are there of arranging the sixteen black or white pieces of a

standard international chess set on the first two rows of the board?

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

its pair.

Answer

6,48,64,800 ways

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

ways.

(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! ways.

• Similarly there are 2 identical rooks, 2 identical knights and 2 identical

bishops. Each can be arranged in 2P2 = 2! ways.

Hence, the require answer is

= (16!) / (8! * 2! * 2! * 2!)

= 6,48,64,800