GATE ME 2024 Set 1 — Question 7
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General Aptitude → Quantitative Aptitude → Permutations & Combinations
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Question
How many combinations of non-null sets are possible from the subsets of satisfying the conditions: (i) is a subset of , and (ii) is a subset of ?
- A.28
- B.27
- C.18
- D.19
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Correct answer
(MTA)
Solution
We are looking for the number of triples such that .
In such problems, the phrase 'from the subsets of ' typically implies that the largest set in the chain is itself, i.e., .
Thus, we need to find the number of pairs such that .
For each element , there are 3 possibilities for its placement in the chain :
However, we must subtract the cases where is the null set (). If , then we are just counting the number of subsets of , which is .
Therefore, the number of valid combinations is .
In such problems, the phrase 'from the subsets of ' typically implies that the largest set in the chain is itself, i.e., .
Thus, we need to find the number of pairs such that .
For each element , there are 3 possibilities for its placement in the chain :
1. (which implies )
2. but
3.
Total number of such pairs is .However, we must subtract the cases where is the null set (). If , then we are just counting the number of subsets of , which is .
Therefore, the number of valid combinations is .
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