Hello, sssouljah!

There are: 2 Kings, 2 Queens, 4 Bishops. 4 Knights, 4 Rooks, and 16 Pawns.In a game of chess there are 32 pieces, including 4 Rooks.

If all the pieces were placed in a line in random order,

what is the probability that no rooks are next to each other?

For simplicity, assume that all 32 pieces are distinguishable.

. . Then there are: . possible arrangements.

Place the 28 non-Rooks in row, leaving a space before, after and between them.

. .

There are 28 pieces and 29 spaces.

The 28 pieces can be arranged in ways.

Select 4 spaces for the Rooks.

. . The Rooks can be placed in: . ways

Hence, there are: . ways to have the Rooks non-adjacent.

Therefore: .

The answer comes out to: .