(i) Find the number of reatangles of any size in a chess board
(ii) Find the number of squares of any size in a rectangle of size `8xx 5` .

Explanation

A chess board is a sqaure of size `8xx3`
`therefore ` number of rectangles of any size =`underset(r=1)overset(8)sum r^(3)=((8(8+1))/(2))^(2)=36^(2)=1296`
(ii) The number of squares of any size in a reactangle of size`8xx5`
`=underset(r=1)overset(5)sum (9-r)(6-5)=underset(r=1)overset(5)sum (54-15r+r^(2))`
`=54xx5-15(1+2+3+4+5)+(1^(2)+2^(2)+3^(2)+4^(2)+5^(2))`
`=270-225+55=100`