Question

The 9 horizontal and 9 vertical lines on an 8 $\times$ 8 chess-board form r rectangles and s squares. Then, the ratio s : r in its lowest terms is
A. $\dfrac{1}{6}$
B. $\dfrac{{17}}{{108}}$
C. $\dfrac{4}{{27}}$
D.None of the above

Answer 1

Choose rectangles using combination and find the number of rectangles. Find the number of squares using the formula of sum of squares of number and then finally find the ratio s : r.

Complete step-by-step answer:
Firstly, we have to select any two horizontal and vertical lines out of 9 horizontal lines and vertical lines respectively for making rectangle or square. For choosing any two lines we have to use a combination.
$\therefore$Number of rectangles (r) formed are ${}^9{C_2} \times {}^9{C_2}$
$= \dfrac{{9!}}{{2! \times 7!}} \times \dfrac{{9!}}{{2! \times 7!}}$
$= \dfrac{{9 \times 8}}{2} \times \dfrac{{9 \times 8}}{2}$
$= 36 \times 36$
$\therefore$ r $= 36 \times 36$
Now, for squares, there will be 8 $\times$ 8 = 64 squares of size 1 $\times$ 1, 7 $\times$ 7 = 49 squares of size 2 $\times$ 2 and so on.
Thus, number of squares will be ${8^2} + {7^2} + {6^2} + {5^2} + {4^2} + {3^2} + {2^2} + {1^2}$ .
Sum of squares of numbers is $\dfrac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}$ i.e. $\dfrac{{8\left( {8 + 1} \right)\left( {16 + 1} \right)}}{6} = 12 \times 17$
$\therefore$s $= 12 \times 17$
Now, taking the ratio of s : r
$= \dfrac{{12 \times 17}}{{36 \times 36}}$
$= \dfrac{{17}}{{108}}$
Thus, the ratio s : r is equal to $17:108$ .
Hence correct answer is B.

Note:

Size of square	Numbers of squares
1 $\times$ 1	8 $\times$ 8 = 64
2 $\times$ 2	7 $\times$ 7 = 49
3 $\times$ 3	6 $\times$ 6 = 36
4 $\times$ 4	5 $\times$ 5 = 25
5 $\times$ 5	4 $\times$ 4 = 16
6 $\times$ 6	3 $\times$ 3 =9
7 $\times$ 7	2 $\times$ 2 = 4
8 $\times$ 8	1 $\times$ 1 = 1