Question 7 At the beginning of a game of chess, the back row consists of two (identical) rooks on the corners, then two knights next to them, then two bishops, and finally a king and a queen in the center. Bobby Fischer, an American chess legend, popularized the idea of randomly shuffled starting positions. In how many distinct ways can the eight pieces above be aligned? (You do not need to worry about color of the squares, or any other practical considerations for the actual game!)
Solution-
According to question
In the game of chess, the back row consists of two (identical) rooks on the corners, then two knights next to them, then two bishops, and finally a king and a queen in the center.
If one randomly shuffled starting positions then number of distinct ways for the eight pieces above can be aligned are
= Permutation of 8 out of 8 divided by (2!×2!×2!)
(because are 3 pairs identical)
= (8!)/(2!×2!×2!)
=(8×7×6×5×4×3×2×1)/(2×1×2×1×2×1)
= 40320/8
=5040
Hence, there are 5040 ways of aligning the starting positions.
Question 7 At the beginning of a game of chess, the back row consists of two...
At the beginning of a game of chess, the back row consists of two (identical) rooks on the corners, then two knights next to them, then two bishops, and finally a king and a queen in the center. If the pieces are randomly shuffled in starting positions, In how many distinct ways can the eight pieces above be aligned? (You do not need to worry about color of the squares, or any other practical considerations for the actual game!)