Question

The number of rectangles excluding squares from a rectangle of sizes 9\[ \times \]6 is
A. 391
B. 791
C. 842
D. 250

Answer 1

Hint: Number the lines from 1 to 10 and 1 to 7 to count the rectangles. Then count the total number of squares one by one. After that, we will count the number of total rectangles excluding the squares.

Complete step by step solution: If you are given a grid of size 9\[ \times \]6 then the number of rectangles means you have to first choose two horizontal lines and two vertical lines from all the line.
As we can see the total number of vertical and horizontal lines is 7 and 10 respectively.

To make a rectangle, we need any two lines among vertical lines and any two from the horizontal ones,
Now, total number of rectangle are,
 \[ = {}^{{\text{10}}}{{\text{C}}_{\text{2}}}{ \times }{}^{\text{7}}{{\text{C}}_{\text{2}}}\]
On using the formula, \[{}^{\text{n}}{{\text{C}}_{\text{r}}}{\text{ = }}\dfrac{{{\text{n!}}}}{{{\text{(n - r)!r!}}}}\], we get,
\[ = \dfrac{{10!}}{{2!\left( {10 - 2} \right)!}} \times \dfrac{{7!}}{{2!(7 - 2)!}}\]
On further simplification we get,
= \[\dfrac{{10 \times 9 \times 8!}}{{2!8!}} \times \dfrac{{7 \times 6 \times 5!}}{{2!5!}}\]
\[ = 45 \times 21\]
\[ = 945\]

Now if we try to calculate the number of the squares,
We have 1 unit size boxes, with the number of (9\[ \times \]6) boxes.
We have 2 unit size boxes, with the number of (8\[ \times \]5) boxes.
We have 3 unit size boxes, with the number of (7\[ \times \]4) boxes.
We have 4 unit size boxes, with the number of (6\[ \times \]3) boxes.
We have 5 unit size boxes, with the number of (5\[ \times \]2) boxes.
We have 6 unit size boxes, with the number of (4\[ \times \]1) boxes.

So, total number of squares \[ = {\text{ }}(9 \times 6){\text{ }} + {\text{ }}(8 \times 5){\text{ }} + {\text{ }}(7 \times 4){\text{ }} + {\text{ }}(6 \times 3){\text{ }} + {\text{ }}(5 \times 2){\text{ }} + {\text{ }}(4 \times 1)\]\[ = 54 + 40 + 28 + 18 + 10 + 4\] \[ = 154\]
Now, the number of rectangles excluding squares, \[ = 945 - 154\] \[ = 791\].

Hence the correct option is (B).

Note: While counting the number of squares we get n – 1 types of unit boxes if we get m horizontal lines and n vertical lines and \[m{\text{ }} > {\text{ }}n.{\text{ }}If{\text{ }}n{\text{ }} > {\text{ }}m\] then we have m – 1 types of unit boxes.