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_____
Correct Answer: 214
Solution and Explanation
Step 1: Define the set E
We are given that $E$ is the set of numbers that are either multiples of 3 or multiples of 7. This means we are interested in counting the elements that are divisible by 3, divisible by 7, and avoiding double-counting those that are divisible by both 3 and 7 (i.e., divisible by 21).
Step 2: Set notation for n(E)
The number of elements in $E$ can be calculated using the principle of inclusion-exclusion:
$$ n(E) = \text{(multiples of 3)} + \text{(multiples of 7)} - \text{(multiples of 21)} $$
This formula is derived to avoid double-counting the multiples of both 3 and 7.
Step 3: Count the multiples of 3
The multiples of 3 form the sequence $3, 6, 9, \dots, 1998$. To find how many terms are in this sequence, we divide the largest multiple of 3 within the range (1998) by 3:
$$ \frac{1998}{3} = 666. $$
Thus, there are 666 multiples of 3.
Step 4: Count the multiples of 7
The multiples of 7 form the sequence $7, 14, 21, \dots, 1995$. To find how many terms are in this sequence, we divide the largest multiple of 7 within the range (1995) by 7:
$$ \frac{1995}{7} = 285. $$
Thus, there are 285 multiples of 7.
Step 5: Count the multiples of both 3 and 7 (multiples of 21)
The multiples of 21 (since $21 = 3 \times 7$) form the sequence $21, 42, 63, \dots, 1995$. To find how many terms are in this sequence, we divide the largest multiple of 21 within the range (1995) by 21:
$$ \frac{1995}{21} = 95. $$
Thus, there are 95 multiples of 21.
Step 6: Apply the inclusion-exclusion principle
Now, we can apply the inclusion-exclusion principle to find the total number of elements in $E$:
$$ n(E) = 666 + 285 - 95 = 856. $$
Thus, the total number of elements in $E$ is 856.
Step 7: Calculate the probability P(E)
The total number of possible outcomes (in this case, the total number of elements under consideration) is 2000. Therefore, the probability of selecting an element from $E$ is:
$$ P(E) = \frac{856}{2000}. $$
Step 8: Multiply the probability by 500
We are asked to find $P(E) \times 500$. Using the probability we calculated:
$$ P(E) \times 500 = \frac{856}{2000} \times 500 = \frac{856 \times 500}{2000} = \frac{856}{4} = 214. $$
Thus, the final result is 214.
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