Power Sets
Consider a set S having exactly n elements; the number of subsets of S that contain at least two elements is given by which expression in terms of n?
Select the correct option:
Solution
2n−n−1
The total number of subsets of an n-element set is 2^n, since each element is independently included or excluded, a core counting theorem for power sets. Subsets with fewer than two elements are the empty set (1 subset) and the singletons (n subsets). Subtracting these unwanted cases gives 2^n - n - 1 subsets containing at least two elements. This complementary counting approach is a recurring JEE Advanced strategy: count the total and remove forbidden configurations. Option 2^n - n forgets to remove the empty set. Option 2^n - 1 removes only the empty set, leaving all singletons incorrectly included. Option 2^(n-1) counts subsets of even size or half the power set, unrelated to this constraint. Hence the required count is 2^n - n - 1. Plausibility check: for n = 3 the formula gives 8 - 3 - 1 = 4, matching the explicit subsets {a,b},{b,c},{a,c},{a,b,c}, confirming correctness for a small concrete case.
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About This Question
- Subject
- mathematics
- Chapter
- sets, relations and functions
- Topic
- power sets
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
2n−n−1
The total number of subsets of an n-element set is 2^n, since each element is independently included or excluded, a core counting theorem for power sets. Subsets with fewer than two elements are the empty set (1 subset) and the singletons (n subsets). Subtracting these unwanted cases gives 2^n - n - 1 subsets containing at least two elements. This complementary counting approach is a recurring JEE Advanced strategy: count the total and remove forbidden configurations. Option 2^n - n forgets to remove the empty set. Option 2^n - 1 removes only the empty set, leaving all singletons incorrectly included. Option 2^(n-1) counts subsets of even size or half the power set, unrelated to this constraint. Hence the required count is 2^n - n - 1. Plausibility check: for n = 3 the formula gives 8 - 3 - 1 = 4, matching the explicit subsets {a,b},{b,c},{a,c},{a,b,c}, confirming correctness for a small concrete case.
This easy difficulty mathematics question is from the chapter sets, relations and functions, covering the topic of power sets. It appeared in the 2025 exam.
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