Key Concepts and Formulas

• Principle of Inclusion-Exclusion for Two Events: To find the number of elements in the union of two sets A and B, we use the formula: $N(A \cup B) = N(A) + N(B) - N(A \cap B)$ This principle ensures that elements belonging to both sets are counted exactly once.
• Counting Multiples: The number of multiples of an integer $k$ within a range $1, \ldots, N$ is given by the floor function: $\lfloor N/k \rfloor$. The floor function rounds down to the nearest integer.
• Least Common Multiple (LCM): An integer is a multiple of both $x$ and $y$ if and only if it is a multiple of their Least Common Multiple, denoted as $\text{LCM}(x, y)$. This concept is crucial for finding the number of elements in the intersection of sets of multiples.

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Step-by-Step Solution

1. Define the Sample Space and Total Outcomes The problem states that an integer is chosen at random from the integers $1, 2, 3, \ldots, 50$. This set of integers forms our sample space. The total number of possible outcomes (integers) is $N(S) = 50$. Each integer has an equal chance of being chosen.

2. Define the Events We are looking for the probability that the chosen integer is a multiple of at least one of 4, 6, and 7. For this type of problem, when determining the number of favorable outcomes for "at least one" condition, it is often most effective to focus on the numbers that share common factors, as these frequently form the core of the union. Let's define the primary events for multiples of 4 and 6:

   • Event A: The chosen integer is a multiple of 4.
   • Event B: The chosen integer is a multiple of 6.

3. Calculate the Number of Outcomes for Individual Events To find the number of multiples of an integer $k$ within the range $1, \ldots, 50$, we use the floor function $\lfloor 50/k \rfloor$.

   • For Event A (multiples of 4): We need to find the count of numbers in $\{1, \ldots, 50\}$ that are multiples of 4. $N(A) = \lfloor 50/4 \rfloor = \lfloor 12.5 \rfloor = 12$. (These multiples are $4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48$)

   • For Event B (multiples of 6): We need to find the count of numbers in $\{1, \ldots, 50\}$ that are multiples of 6. $N(B) = \lfloor 50/6 \rfloor = \lfloor 8.33\ldots \rfloor = 8$. (These multiples are $6, 12, 18, 24, 30, 36, 42, 48$)

4. Calculate the Number of Outcomes for the Intersection of Events An integer is a multiple of both $x$ and $y$ if and only if it is a multiple of their Least Common Multiple (LCM).

   • For Event A $\cap$ B (multiples of 4 and 6): These are numbers that are multiples of both 4 and 6. So, we need to find multiples of LCM(4, 6). $\text{LCM}(4, 6) = 12$. $N(A \cap B) = \lfloor 50/12 \rfloor = \lfloor 4.16\ldots \rfloor = 4$. (These multiples are $12, 24, 36, 48$)

5. Apply the Principle of Inclusion-Exclusion Now we substitute all the calculated counts into the PIE formula for two events to find the number of integers that are multiples of at least one of 4 or 6: $N(A \cup B) = N(A) + N(B) - N(A \cap B)$ $N(A \cup B) = 12 + 8 - 4$ $N(A \cup B) = 20 - 4$ $N(A \cup B) = 16$ So, there are 16 integers between 1 and 50 (inclusive) that are multiples of at least one of 4 or 6.

6. Calculate the Probability The probability is the number of favorable outcomes divided by the total number of outcomes. $P(A \cup B) = \frac{N(A \cup B)}{N(S)}$ $P(A \cup B) = \frac{16}{50}$ $P(A \cup B) = \frac{8}{25}$

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Common Mistakes & Tips

• Don't forget LCM: Always use the Least Common Multiple (LCM) when finding the count of numbers that are multiples of two or more integers (i.e., for intersections of events). A common mistake is to use the product of the numbers instead of their LCM, which is only correct if the numbers are coprime.
• Correct use of Floor Function: Ensure you correctly apply the floor function ($\lfloor \cdot \rfloor$) when counting multiples. It always rounds down to the nearest integer, which accurately reflects the count of full groups of multiples.
• Careful with Arithmetic: The Inclusion-Exclusion Principle involves several additions and subtractions. Double-check your calculations to avoid simple arithmetic errors that can lead to an incorrect final answer.

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Summary

This problem demonstrates the application of the Principle of Inclusion-Exclusion in probability. The key steps involved defining the sample space and events, systematically calculating the number of elements in individual sets and their intersections using the concept of LCM and the floor function, and then applying the PIE formula to find the number of favorable outcomes. Finally, the probability is calculated by dividing the number of favorable outcomes by the total number of outcomes. By focusing on the primary interactions between multiples of 4 and 6, we find that 16 integers satisfy the condition within the given range.

The final answer is $\boxed{\frac{8}{25}}$, which corresponds to option (A).