GATE CSE 2016 Set 2 | Question: GA-09

Question

In a $2 \times 4$ rectangle grid shown below, each cell is rectangle. How many rectangles can be observed in the grid?

$$\begin{array}{|c|c|c|c|c|}\hline{\;\;\;}&{\;\;\;}&{\;\;\;}&{\;\;\;}\\\hline{}&{}&{}&\\\hline\end{array}$$

• A. $21$
• B. $27$
• C. $30$
• D. $36$

Comments

How is rectangle of size 3/5/7 possible ?

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Size 5 and 7 not possible. For size 3 look row 1  three horizontal rectangles. Total there are four such triangles of size three.

Answer 1

To form a rectangle, we must choose two horizontal sides and two vertical sides. Since there are three horizontal lines,
We can choose the horizontal sides in ${^3}C{_2}$ ways.
Similarly, to choose $2$ vertical lines out of $5$ vertical lines is ${^5}C{_2}.$

So, answer is ${5\choose 2}\times {3\choose 2}.$

Correct Answer: $C$

Answer 2

Here answer is (C)  30.

Rectangle of Size 1 => 8

Size 2 => 10

Size 3 => 4

Size 4 => 5

Size 5 => 0

Size 6 => 2

Size 7 => 0

Size 8 => 1

Comments

How is rectangle of size 3/5/7 possible ?

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make rectangles using 3 unit cells @swati

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Got it thanks.

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This includes squares also....How to find only rectangles??

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@nivedha

every square is also a rectangle!

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Every square is a rectangle, but reverse need not be true.

$\text{Number of rectangles} = \:\: ^{\text{Horizontal}}\:C_{2}\times ^{\text{Vertical}}C_{2}$

Answer 3

3C2 X 5C2 = 30

Answer 4

5C2 * 3C2