Question:

Step 1: Determine the total number of elements. The set is $\{1, 2, \ldots, 2000\}$. The total number of elements is $N = 2000$. Step 2: Calculate the number of multiples of 3. The number of multiples of 3 in the set is given by: $$N_3 = \left\lfloor \frac{2000}{3} \right\rfloor = 666$$ Step 3: Calculate the number of multiples of 7. The number of multiples of 7 in the set is given by: $$N_7 = \left\lfloor \frac{2000}{7} \right\rfloor = 285$$ Step 4: Calculate the number of multiples of both 3 and 7. Numbers that are multiples of both 3 and 7 are multiples of their least common multiple, which is $\text{lcm}(3, 7) = 21$. The number of multiples of 21 in the set is given by: $$N_{21} = \left\lfloor \frac{2000}{21} \right\rfloor = 95$$ Step 5: Calculate the number of multiples of 3 or 7. Using the Principle of Inclusion-Exclusion, the number of elements that are multiples of 3 or 7 is: $$N_{3 \text{ or } 7} = N_3 + N_7 - N_{21}$$ $$N_{3 \text{ or } 7} = 666 + 285 - 95 = 951 - 95 = 856$$ Step 6: Calculate the probability $p$. The probability $p$ that the chosen number is a multiple of 3 or a multiple of 7 is the ratio of the number of favorable outcomes to the total number of outcomes: $$p = \frac{N_{3 \text{ or } 7}}{N} = \frac{856}{2000}$$ Step 7: Calculate $7000p$. $$7000p = 7000 \times \frac{856}{2000}$$ $$7000p = \frac{7000}{2000} \times 856$$ $$7000p = \frac{7}{2} \times 856$$ $$7000p = 7 \times 428$$ $$7000p = 2996$$