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Subsets of A×B with at Least Three Elements | JEE Sets

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Question
Let AA and BB be two sets containing four and two elements respectively. Then the number of subsets of the set A×BA \times B, each having at least three elements, is:
A219219correct
B256256
C275275
D510510
Solution
n(A×B)=4×2=8⇒total subsets=28=256.n(A \times B) = 4 \times 2 = 8 \Rightarrow \text{total subsets} = 2^8 = 256.
 Subtract subsets with fewer than 33 elements:
256−(80)−(81)−(82)=256−1−8−28=219.256 - \binom{8}{0} - \binom{8}{1} - \binom{8}{2} = 256 - 1 - 8 - 28 = 219.
 Correct answer: (1)
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