JEE Advanced 2021 Paper 2 · Q17 · Probability
A number is chosen at random from the set $\{1, 2, 3, \ldots, 2000\}$. Let $p$ be the probability that the chosen number is a multiple of $3$ or a multiple of $7$. Then the value of $500p$ is _____.
Reveal answer + step-by-step solution
Correct answer:214
Solution
Multiples of 3 in $\{1,\ldots,2000\}$: $\lfloor 2000/3\rfloor = 666$. Multiples of 7: $\lfloor 2000/7\rfloor = 285$. Multiples of 21: $\lfloor 2000/21\rfloor = 95$. By inclusion-exclusion: $666 + 285 - 95 = 856$. So $p = \dfrac{856}{2000} = \dfrac{107}{250}$, and $500p = \dfrac{856}{4} = 214$.
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