Question

Consider a Master Set S= {1,2,3,4….12} How many subsets can be formed which will contain one or more elements of S (including all S) such that the elements of the sets are integral multiples of the smallest subset of the set.

• A. 2246
• B. 2824
• C. 3452
• D. 2102
• E. 1857

Answer 1

The correct option is D 2102

If 1 is the smallest element of the set, all or none of the other elements can be selected in $2^{11}$ ways, as for each number from 2 to 12, there are two options, of getting selected or not getting selected. There are 11 such numbers 2-12 including both, therefore 2.2.2…..11 times = $2^{11}$

Similarly, If 2 is the smallest element in the set, 4,6,8,10 and 12 can be selected in 2⁵ ways

If 3 is the smallest element in the set 6,9,12 $2^3$ different sets

Required Solution = $2^{11}+2^5+2^3+2^2+2^1+2^1+6=2102$