If the number of rectangles formed on a chess | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let the size of the chessboard be $n 	imes n$. The number of rectangles on an $n 	imes n$ grid is given by $\binom{n+1}{2} 	imes \binom{n+1}{2}

Vedclass · Accepted Answer

$204$

Vedclass · Answer

(C) Let the size of the chessboard be $n 	imes n$. The number of rectangles on an $n 	imes n$ grid is given by $\binom{n+1}{2} 	imes \binom{n+1}{2}$.Given $\left(\frac{n(n+1)}{2}ight)^2 = 1296 = (36)^2$.Thus,$\frac{n(n+1)}{2} = 36$,which implies $n(n+1) = 72$,so $n = 8$.The total number of squares on an $n 	imes n$ board is $\sum_{k=1}^{n} k^2$.For $n = 8$,the sum is $1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2$.Using the formula $\frac{n(n+1)(2n+1)}{6}$,we get $\frac{8 	imes 9 	imes 17}{6} = 4 	imes 3 	imes 17 = 204$.Therefore,the correct option is $C$.

If the number of rectangles formed on a chessboard is $1296$,then the total number of squares formed on the chessboard is:

Similar Questions

Three of six vertices of a regular hexagon are chosen at random. The probability that the triangle formed by these three vertices is equilateral is

Let $l_1$ and $l_2$ be two lines intersecting at $P$. If $A_1, B_1, C_1$ are points on $l_1$,and $A_2, B_2, C_2, D_2, E_2$ are points on $l_2$,and if none of these points coincides with $P$,then the number of triangles formed by these eight points is:

Let $P_{1}, P_{2}, \ldots, P_{15}$ be $15$ points on a circle. The number of distinct triangles formed by points $P_{i}, P_{j}, P_{k}$ such that $i+j+k \neq 15$ is:

How many necklaces can be formed by selecting $4$ beads from $6$ different colored glass beads and $4$ beads from $5$ different colored metal beads?

In how many ways can $5$ boys and $3$ girls sit in a row so that no two girls are together?

Vedclass Test Series

Exam Paper Generator

Online Exam Module