Question

If three squares are chosen on a chess board, the chance that they should be in a diagonal line is

• A. $\frac{7}{744}$
• B. $\frac{5}{744}$
• C. $\frac{17}{744}$
• D. $\frac{11}{744}$

Answer 1

The correct option is A

$\frac{7}{744}$

Explanation for the correct option.

Step 1. Find the total number of outcomes.

There are a total of $64$ squares on a chess board.

When selecting any $3$ squares the total number of outcomes is given as: ${}^{64}C_3$.

The value of the term ${}^{64}C_3$ can be found using ${}^{n}C_r=\frac{n!}{r!(n-r)!}$ as:

$\begin{array}{rcl}{}^{64}C_3& =& \frac{64!}{3!(64-3)!}\\ & =& \frac{64·63·62·61!}{6·61!}\\ & =& 41664\end{array}$

So the total number of outcomes is $41664$.

Step 2. Find the possible number outcomes.

In a chess board there are a total of $26$ diagonal lines: diagonals containing number of squares $2,4,5,6,7$ are each $4$ and there are $2$ main diagonals containing $8$ squares.

$3$ squares cannot be chosen from a diagonal containing $2$ squares.

So from the other $22$ diagonals the chance of picking $3$ squares which are in a diagonal is given as: $4\left({}^3{C}_3+{}^4{C}_3+{}^5{C}_3+{}^6{C}_3+{}^7{C}_3\right)+2{}^8{C}_3$.

Using ${}^{n}C_r=\frac{n!}{r!(n-r)!}$ the value can be found as:

$\begin{array}{rcl}4\left({}^3{C}_3+{}^4{C}_3+{}^5{C}_3+{}^6{C}_3+{}^7{C}_3\right)+2{}^8{C}_3& =& 4\left(\frac{3!}{3!(3-3)!}+\frac{4!}{3!(4-3)!}+\frac{5!}{3!(5-3)!}+\frac{6!}{3!(6-3)!}+\frac{7!}{3!(7-3)!}\right)+2\frac{8!}{3!(8-3)!}\\ & =& 4\left(1+\frac{4\times 3!}{3!}+\frac{5\times 4\times 3!}{3!\times 2}+\frac{6\times 5\times 4\times 3!}{6\times 3!}+\frac{7\times 6\times 5\times 4!}{6\times 4!}\right)+2\frac{8\times 7\times 6\times 5!}{6\times 5!}\\ & =& 4\left(1+4+10+20+35\right)+2\times 56\\ & =& 4\times 70+112\\ & =& 280+112\\ & =& 392\end{array}$

Thus the number of favorable outcomes is $392$.

Step 3. Find the probability.

The probability that the three squares lie in a diagonal line is given by the ratio of number of favorable outcomes to the total number outcomes. So the probability is:

$\begin{array}{rcl}P& =& \frac{392}{41664}\\ & =& \frac{7}{744}\end{array}$

If three squares are chosen on a chess board, the chance that they should be in a diagonal line is $\frac{7}{744}$.

Hence, the correct option is (A) .