Question

In the given figure is a 10 by 10 grid. By using concept of permutations and combinations find the number of possible rectangles in the figure.​

Answer 1

$\large\underline{\sf{Solution-}}$

As it is provided a grid of 10 by 10

It means there are 11 vertical lines and 11 horizontal lines.

In order to form a rectangle of any size, we need two vertical lines and 2 horizontal lines.

Now, we have given 11 vertical lines.

So, Number of ways in which 2 vertical lines can be selected from 11 vwrtical lines is

$\rm \:  =  \:  \: ^{11}C_2$

$\rm \:  =  \:  \: \dfrac{11!}{2! \: (11 - 2)!}$

$\rm \:  =  \:  \: \dfrac{11!}{2! \: 9!}$

$\rm \:  =  \:  \: \dfrac{11 \times 10 \times 9!}{2 \times 1 \times \: 9!}$

$\rm \:  =  \:  \: 55$

Now, we have 11 horizontal lines.

So,

Number of ways in which 2 horizontal lines can be selected from 11 horizontal lines is

$\rm \:  =  \:  \: ^{11}C_2$

$\rm \:  =  \:  \: \dfrac{11!}{2! \: (11 - 2)!}$

$\rm \:  =  \:  \: \dfrac{11!}{2! \: 9!}$

$\rm \:  =  \:  \: \dfrac{11 \times 10 \times 9!}{2 \times 1 \times \: 9!}$

$\rm \:  =  \:  \: 55$

Hence,

Total number of possible rectangles in the figure is

$\rm \:  =  \:  \: ^{11}C_2 \: \times \: ^{11}C_2$

$\rm \:  =  \:  \: 55 \times 55$

$\rm \:  =  \:  \: 3025$

Formula Used :-

$\boxed{ \rm{ \: ^{n}C_r \: = \: \frac{n!}{r! \: ( \: n \: - \: r \: )!}}}$

Additional Information :-

Short cut tricks

If we have a grid of n × n, then

$\boxed{ \rm{ Number \: of \: squares \: = \: \sum_{r = 1}^n \: {r}^{2} = \dfrac{n(n + 1)(2n + 1)}{6} }}$

$\boxed{ \rm{ Number \: of \: rectangles \: = \: \sum_{r = 1}^n \: {r}^{3} = \dfrac{ {n}^{2} {(n + 1)}^{2} }{4} }}$