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Question

How many rectangles does a 4×4 square has?

A
13+23+33+43
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B
13+23+33
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C
13+23+33+43+53
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D
13+23+33+43+53+63
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Solution

The correct option is A 13+23+33+43

Along the top row of the 4×4 square,
Total number of rectangle of size 1×1=4
Total number of rectangle of size 1×2=3
Total number of rectangle of size 1×3=2
Total number of rectangle of size 1×4=1
Total Number of rectangles in first row = 10
Now, along the first column,
Total number of rectangle of size 1×1=4
Total number of rectangle of size 1×2=3
Total number of rectangle of size 1×3=2
Total number of rectangle of size 1×4=1
Total Number of rectangles in first column = 10
Hence the total number of rectangle in 4×4 square grid =10×10=102
Now 102=13+23+33+43
Therefore, total no of rectangles in 4×4 square grid =13+23+33+43

Alternative Solution:

We know that n×n square has 13+23+33+.....+n3 rectangles
Similarly, 4×4 square grid has 13+23+33+43 rectangles.

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