Question

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!)

Answer 1

The number of distinct arrangments or alignments of 'n' objects such that within n objects, there are r₁ identical objects of one kind, r₂ identical objects of another kind, and r₃ identical objects of a third kind, is given by the formula

Number of arrangements = $n!/r_1!r_2!r_3!$

Here, there are a total of 8 pieces(Two rooks, two knights, two bishops, and a king and queen). This implies that

$n=8$.

Also, there are 2 rooks, so $r_1=2$

There are two knights, so $r_2=2$

There are two bishops,so $r_3=2$

Applying are formula, we get the number of distinct alignments as

$=8*7*6*5*4*3*2*1/2*1*2*1*2*1$

$=8*7*6*5*4*3*2/2*2*2$

Hence, the pieces can be aligned in 5040 ways