JEE Main 2021 · 24 February · evening · Single correct · +4 / −1
The probability that two randomly selected subsets of the set {1, 2, 3, 4, 5} have exactly two elements in their intersection, is :
- (a) 135/2⁹
- (b) 65/2⁸
- (c) 65/2⁷
- (d) 35/2⁷
Answer and solution
Answer (a) 135/2⁹
Solution
Maths sir · 3 steps
Do not list subsets. Decide the fate of each of the 5 numbers one at a time.
- Total outcomes. Each of the 5 numbers has 4 choices: in A only, in B only, in both, or in neither. So the number of ordered pairs (A, B) is 4⁵ = 1024.
- Favourable outcomes. Pick the 2 numbers that go in both sets in C(5, 2) = 10 ways. Each of the other 3 numbers must stay out of the intersection, so it has 3 choices: A only, B only, or neither. That gives 3³ = 27 ways.
P = (C(5, 2) × 3³)/4⁵ = (10 × 27)/1024 = 270/1024
- So. 270/1024 = 135/512 = 135/2⁹, which is option (a).
The idea to keep. For questions on pairs of subsets, give each element its own choices: in A only, in B only, in both, or in neither. Then multiply. This turns subset counting into a simple product.
Answer from the NTA final answer key. Solution by PrepWiser.