Question

Find the total number of rectangle possible in the given L'shaped grid.

Answer is 900, provide explanation and solution.​

Answer 1

Answer:

$\huge \pink{900}$

Step-by-step explanation:

In a simple rectangle of m × n dimension , total number of rectangles are

$= \binom{m + 1}{2} . \binom{n + 1}{2}$

Think of this like choosing vertexes of rectangles from m + 1 and n + 1 options ( count number of lines they will be one more then boxes ).

Now First take 10 × 4 vertical portion , total no. of rectangles in it will be

$= \binom{11}{2} . \binom{5}{2}$

Now take 9 × 4 horizontal portion , rectangles in it are

$= \binom{10}{2}. \binom{5}{2}$

But note that rectangles those are in common area of 4 × 4 has been counted twice , such rectangles are

$= \binom{5}{2} . \binom{5}{2}$

So we will minus these rectangles I.e. require

number of rectangles will be

$\binom{11}{2} . \binom{5}{2} + \binom{10}{2} . \binom{5}{2} - \binom{5}{2} . \binom{5}{2} \\ \\ = 55 \times 10 + 45 \times 10 + 10 \times 10 \\ \\ \\ = (55 + 45 - 10)10 = 900$

Answer 2

Answer:

$\large\underline \red{\underline{\bigstar{\textbf{\textsf{\: solution\::-}}}}}$

here we need to find total number of rectangle possible in the given l - shaped grid

so for finding this we need to use the total number of rectangle formula

$\sf \rightarrow total \: number \: of \: rectangle = \binom{h}{2} \binom{v}{2} = \frac{h(h - 1)v(v - 1)}{4}$

here, h is horizontal line

and v is vertical line

now according to question :

suppose you take distinct horizontal lines and two distinct vertical line

not every pair of 2 horizontal line and 2 vertical lines will bound a rectangle

so we have to count a bit more carefully

now in blue grid, in just this gird every pair of 2 horizontal line and 2 vertical lines will from a rectangle

now the blue And yellow grids overlap in a green gird (5 by 5) lines

so the rectangle in this grid will be double counted to compensate we can subtract the rectangle just in their green grid

now by using formula we get :-

$\sf \rightarrow \: blue \: grid = \binom{11}{2} \binom{5}{2} = \frac{11(10)5(4)}{2} = 550$

$\sf \rightarrow \: yellow \: grid = \binom{5}{2} \binom{10}{2} = \frac{5(4)10(9)}{4}$

$\sf \rightarrow \: green \: grid = \binom{5}{2} \binom{5}{2} = \frac{5(4)5(4)}{4} = 100$

let's count the number of rectangle by adding and subtracting the blue, green, yellow grid

$\sf \rightarrow \: 500 + 450 - 100 = 900$

$\sf \rightarrow \: 900$