GATE Overflow Test Series | Discrete Mathematics | Test 2 | Question: 16

Question

A board is formed by combining $100$ squares of same size, $10$ in each direction. The total number of squares (of all sizes) that can be formed from the resultant board is $\_\_\_\_$

Answer 1

Number of squares of size $1 = 10^2 = 100$

Number of squares of size $2 = 9^2 = 81$

Number of squares of size $3 = 8^2 = 64$

Number of squares of size $4 = 7^2 = 49$

Number of squares of size $5 = 6^2 = 36$

Number of squares of size $6 = 5^2 = 25$

Number of squares of size $7 = 4^2 = 16$

Number of squares of size $8 = 3^2 = 9$

Number of squares of size $9 = 2^2 = 4$

Number of squares of size $10 = 1^2 = 1$

Total number of squares $ =$ Sum of squares till $10$

$\qquad= \dfrac{n(n+1)(2n+1)}{6}$

$\qquad =\dfrac{10.11.21}{6}=385$

Comments

Sir, if we are forming 100 rectangles can we write it as (11 C 2) x (11 C 2)?

---

No ,then ans is (101 C 2)*(101 C 2 ) this is total no of rectangles possible in 10*10 Grid.