# Thread: Probability Question

1. ## Probability Question

In a game of chess there are 32 pieces (16 black and 16 white). Out of the 32 pieces there are 4 rooks, what is the chance that if all the pieces were placed in a line, no rooks are next to each other? Pieces are selected at random

Regards

Sssouljah

2. Hello, sssouljah!

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?
There are: 2 Kings, 2 Queens, 4 Bishops. 4 Knights, 4 Rooks, and 16 Pawns.

For simplicity, assume that all 32 pieces are distinguishable.
. . Then there are: .$\displaystyle 32!$ possible arrangements.

Place the 28 non-Rooks in row, leaving a space before, after and between them.
. . $\displaystyle \_\ X\ \_\ X\ \_\ X\ \_\;\; \hdots\;\; \_\ X\ \_\ X\ \_$

There are 28 pieces and 29 spaces.

The 28 pieces can be arranged in $\displaystyle 28!$ ways.

Select 4 spaces for the Rooks.
. . The Rooks can be placed in: .$\displaystyle _{29}P_4$ ways

Hence, there are: .$\displaystyle (28!)(_{29}P_4)$ ways to have the Rooks non-adjacent.

Therefore: .$\displaystyle \boxed{P(\text{no adjacent Rooks}) \;=\;\frac{(28!)\left(_{29}P_4\right)}{32!}}$

The answer comes out to: .$\displaystyle \frac{819}{1240} \;\approx\;0.66$