Question

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 $500\, p$ is_____

Answer 1

Let $A =\{1,2,3, \ldots ., 2000\}$
Let $E _{1}=3 m , 1 \leq m \leq 666, m \in N$
$ E _{2}=7 k , 1 \leq k \leq 285, k \in N$
$= n \left( E _{1} \cup E _{2}\right)=666+285-95=856$
$= P \left( E _{1} \cup E _{2}\right)=856 / 2000$
$=500 \beta=856 / 4=214$