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Q. Determine the number of subsets of {3, 7, 9, 11, 24} with the property that the sum of the elements in the subset is less than 28.

My approach: First I calculated the total number of subsets that is $2^\left | S \right |$ , which in this case is $2^5$=32.

Then from the total number of subsets I subtract those subsets which give sum $\geq $ 28, to get the subset whose sum $\geq $ 28 , I include 24 in each of the subsets $\left \{24, \texttt{_,_,_,_ } \right \}$ so by filling the rest of the remaining four places will give $2^4$ subsets, these $2^4$ subsets has sum $\geq $ 28, so we subtract these subsets from the total number of subsets to get the subsets that has sum $\leq $ 28 that is $2^5$- $2^4= 16$.

But in the $2^4$ subsets that we have calculated includes a subset $\left \{24, \texttt{3} \right \}$ whose sum $\leq $ 28 , so we subtract one from the $2^4$ subsets, which is $2^4-1=15$ is the number of subsets whose sum $\geq $ 28

So we get $2^5-(2^4-1)=17 $ as the answer which is the total number of subsets whose sum $\leq $ 28

Is this approach correct to solve these type of questions?

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    Your approach seems ok, but you are actually missing a subset whose sum $\geq$ 28. Hint, it doesn't include 24. – 2017-01-16
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    Ya i got it the subset {3,7,9,11} but if we add 1 to $2^{4}-1$ , the answer we will get is 16. But the answer given in Rosen is 17, What Should I do? – 2017-01-16
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    @laura check my answer now, it should make things clearer – 2017-01-16
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    Thanks for the answer, Ya I missed {24} this would add 1 to the answer, finally giving 17. Thanks once again! – 2017-01-16

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Your approach is correct but requires two slight modifications:

  • When counting the $2^4$ subsets you are subtracting, you are also subtracting the subset $\{24\}$ whose sum does not exceed $28$;

  • There is one other subset that exceeds $28$ which is the one with all the numbers but $24$;

Making these adjustments should be enough to get you to the final, correct answer.

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    Is this sort of approach a good replacement of the tree diagram? – 2017-01-16
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    @laura a tree diagram is just a tool used to keep track of all possible subsets and helps one to not forget about some cases or to avoid double counting, etc... There are always plenty of ways to count the same thing. If you are able to do so without a tree diagram, then go for it! A tree diagram is not more special than your method. – 2017-01-16