Let `A_(1),A_(2)... A_(m)` be non -empty subsets of {1,2,3 ....100}, satisfying the following conditions.
(1) the numbers `|A_(1)|,|A_(2)|....,|A_(m)|` are disjoint.
(2) `A_(1),A_(2) ,... ,A_(m)` are pairwise disjoint.
(Here|A| denotes the number fo elements in the set A.) Then the maximum possible value of m is

Options

A. 13
B. 14
C. 15
D. 16

Answer

a

Explanation

The possibility is
`|A_(1)|=1,|A_(2)|, |A_(3)|=3,........,|A_(m)|=m`lt brgt1+2+3+.....+m`ge100` {because all are disjoint}
`Rightarrow(m(m+1))/2 le100`
`Rightarrow mlt14`
`14^("th") set will have the same size as that of one of the previous sets
So, m=13