Question

The number of the rectangles excluding squares that can be formed from a rectangle of size $9\times 6$ if it is divided by set of parallel lines of unit length as shown in the diagram is

• A. $391$
• B. $791$
• C. $842$
• D. None of these

Answer 1

The correct option is B $791$

Here, $l=9$ and $b=6$
Number of rectangles that can be selected without any restrictions is $={}^7{C}_2\times {}^{10}{C}_2$
Now the squares can be selected of dimensions $r\times r$
$($where $r\in \{1,2,...,6\})$
Now square of dimension $1\times 1$ can be selected in $9\times 6$ ways.
Similarly, square of dimension $2\times 2$ can be selected in $8\times 5$ ways.
Similarly, square of dimension $6\times 6$ can be selected in $4\times 1$ ways.
So the number of squares will be
$\sum _{r=1}^6(10-r)(7-r)$
$\therefore$ Number of required ways $={}^7{C}_2\times {}^{10}{C}_2-\sum _{r=1}^6(10-r)(7-r)$
$=945-\sum _{r=1}^6(70-17r+r^2)$
$=945-(70\times 6-17\times 21+91)=791$