Question

Determine the number of rectangles that can be formed on a chess – board.

Answer 1

Hint:Assume a chess board of $$8\times 8$$. Find the different ways of dimensions possible to determine the number of rectangles.There are 9 horizontal lines and 9 vertical lines. A rectangle will be formed by two vertical and two horizontal lines.And two horizontal lines can be selected in ${}^9{C}_2$ ways.Similarly, number of ways to select 2 lines out of 9 vertical lines is also ${}^9{C}_2$.Thus formulate and find the number of rectangles.

Complete step-by-step answer:
Let us assume that it is a $$8\times 8$$ chessboard. To form a rectangle in the chess board, it can have the following dimension.
$$\begin{array}{rl}& 1\times 1,1\times 2,1\times 3,1\times 4,1\times 5,1\times 6,1\times 7,1\times 8\\ & 2\times 2,2\times 3,2\times 4,2\times 5,2\times 6,2\times 7,2\times 8\\ & 3\times 3,3\times 4,3\times 5,3\times 6,3\times 7,3\times 8\\ & 4\times 4,4\times 5,4\times 6,4\times 7,4\times 8\\ & 5\times 5,5\times 6,5\times 7,5\times 8\\ & 6\times 6,6\times 7,6\times 8\\ & 7\times 7,7\times 8\\ & 8\times 8\end{array}$$
Thus we got 36 ways of how to form the dimension of the rectangle in a chess board. To form a rectangle, we need 4 lines i.e. 2 sets of parallel lines.
We can generalize the total number of rectangles in a $$n\times n$$ chess board to $${}^{n+1}{C}_2\times {}^{n+1}{C}_2$$.
Thus we know that $$n\times n$$ is equal to $$8\times 8$$.
Thus applying n in the above formula,
Number of rectangles $$={}^{8+1}{C}_2\times {}^{8+1}{C}_2={}^9{C}_2\times {}^9{C}_2$$
They are of the form, $${}^n{C}_r={\displaystyle \frac{n!}{r!(n-r)!}}$$.
$${}^9{C}_2={\displaystyle \frac{9!}{2!(9-2)!}}={\displaystyle \frac{9!}{7!2!}}={\displaystyle \frac{9\times 8\times 7!}{7!\times 2\times 1}}={\displaystyle \frac{9\times 8}{2}}=9\times 4=36$$
$$\therefore$$ Number of rectangles $$={}^9{C}_2\times {}^9{C}_2=36\times 36=1296$$.
$$\therefore$$ The number of rectangles that can be formed on a chess board = 1296.

Note: If the chess board is of the dimension $$m\ast n$$, then the total number of rectangles on the $$m\ast n$$ board will be, $${}^{m+1}{C}_2\times {}^{n+1}{C}_2$$, where a rectangle can be formed by selecting 2 lines from (m + 1) and 2 lines from (n + 1).