Let $\{ x,y\}$ be a subset of the set $\{ 1,2,3,\ldots.,9,10\}$. Then the probability that $|x - y|

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Let $\{ x,y\}$ be a subset of the set $\{ 1,2,3,\ldots.,9,10\}$. Then the probability that $|x - y| \leq 5$ is equal to

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Sol. [Key Concept: $\{ x,y\} subseteq\{ 1..10\}$ under the condition: count pairs.]$|x - y| \geq 6$ holds for $(1,7),\ldots.,(1,10),(2,8),(2,9),(2,10),(3,9),(3,10),(4,10)$Number of ways for which $|x - y| \geq 6 = 4 + 3 + 2 + 1 = 10$$$P(|x - y| \leq 5) = 1 - \frac{10}{\ ^{10}C_{2}} = \frac{7}{9}$$

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