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 $\_\_\_\_$.

$S = \{1,2,3,\ldots,2000\},\quad n(S)=2000$

$E: \text{ multiple of } 3 \ \text{ or multiple of } 7$

$p = \dfrac{n(E)}{n(S)} = \dfrac{n(E)}{2000}$

$\text{To find: } 500p$

$E = A\cup B,\quad A=\text{mult. of }3,\ B=\text{mult. of }7$ $1.\ n(A),\ n(B)$ $2.\ n(A\cap B) = n(\text{mult. of }21)$ $3.\ n(A\cup B) = n(A)+n(B)-n(A\cap B) \Rightarrow p$ $\#\{\text{multiples of } d \text{ in } [1,n]\} = \left\lfloor \dfrac{n}{d}\right\rfloor$ $n(A) = \left\lfloor \dfrac{2000}{3}\right\rfloor = 666$ $n(B) = \left\lfloor \dfrac{2000}{7}\right\rfloor = 285$ $n(A\cap B) = n(\text{mult. of }21) = \left\lfloor \dfrac{2000}{21}\right\rfloor = 95$ $n(A\cup B) = 666+285-95 = 856$ $|A\text{ only}|=571,\ |A\cap B|=95,\ |B\text{ only}|=190,\ \text{neither}=1144$ $p = \dfrac{856}{2000} = \dfrac{107}{250}$ $500p = 500\cdot\dfrac{856}{2000} = \dfrac{856}{4} = 214$ $\boxed{500p = 214}$