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Cardinal Number

Let A1, A2 ,....

Question

Let $A_{1}, A_{2}, . . ., A_{m}$ be non-empty subsets of ${1, 2, 3, . . ., 100}$ satisfying the following conditions:
(1) the numbers $| A_{1} |, | A_{2} |, . . ., | A_{m} |$ are distinct ;
(2) $A_{1}, A_{2}, . . ., A_{m}$ are pairwise disjoint.
(Here $| A |$ denotes the number of elements in the set $A$ ).
Then the maximum possible value of $m$ is

13

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14

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15

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16

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Solution

The correct option is A $13$
$The possibility is, | A_{1} | = 1; | A_{2} | = 2; | A_{3} | = 3; . . .; | A_{m} | = m 1 + 2 + 3 + . . . + m \leq 100 (∵ all are disjoint) \Rightarrow \frac{m (m + 1)}{2} \leq 100 \Rightarrow m (m + 1) \leq 200 Since, m \in N \Rightarrow m \leq 13 Max. possible value of m = 13$

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