A number is chosen at random from the set {1, 2, 3, … , 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 ___ .

The sample space consists of the integers from 1 to 2000, so the total number of equally likely outcomes is $$2000$$.

We need the probability that a randomly chosen integer is a multiple of 3 or a multiple of 7.
Let $$A$$ be the set of multiples of 3 and $$B$$ be the set of multiples of 7.

First count the favourable numbers using the principle of inclusion-exclusion.

Number of multiples of 3:
$$|A| = \left\lfloor \frac{2000}{3} \right\rfloor = 666$$

Number of multiples of 7:
$$|B| = \left\lfloor \frac{2000}{7} \right\rfloor = 285$$

Number of multiples of both 3 and 7 (i.e. multiples of $$\operatorname{lcm}(3,7)=21$$):
$$|A \cap B| = \left\lfloor \frac{2000}{21} \right\rfloor = 95$$

Hence the number of integers that are a multiple of 3 or 7 is
$$|A \cup B| = |A| + |B| - |A \cap B| = 666 + 285 - 95 = 856.$$

Therefore the required probability is
$$p = \frac{856}{2000}.$$

Simplify the fraction:
$$\frac{856}{2000} = \frac{107}{250}.$$

Finally, compute $$500p$$:
$$500p = 500 \times \frac{107}{250} = 2 \times 107 = 214.$$

Hence, the value of $$500p$$ is 214.