1. Key Concepts and Formulas

• Probability: The probability of an event $E$ is given by the ratio of the number of favorable outcomes ($n(E)$) to the total number of possible outcomes ($n(S)$), i.e., $P(E) = \frac{n(E)}{n(S)}$.
• Arithmetic Mean - Geometric Mean (AM-GM) Inequality: For any two non-negative real numbers $a$ and $b$, the arithmetic mean is greater than or equal to the geometric mean: $\frac{a+b}{2} \ge \sqrt{ab}$. Equality holds if and only if $a=b$. This is crucial for finding the maximum product given a fixed sum.
• Solving Quadratic Inequalities: Inequalities of the form $(x-a)(x-b) \le 0$ are satisfied for values of $x$ between $a$ and $b$ (inclusive), assuming $a \le b$.

2. Step-by-Step Solution

Step 1: Determine the Sample Space and Total Number of Outcomes ($n(S)$)

• What we are doing: We need to identify all possible pairs of positive integers $(x,y)$ such that their sum is 24. This collection of all possible pairs forms our sample space.
• Why we are doing it: To calculate probability, we first need to know the total number of possible scenarios.
• Math: Let the two positive integers be $x$ and $y$. Given: $x+y=24$, where $x, y \in \{1, 2, 3, \ldots\}$. Since $x \ge 1$ and $y \ge 1$, we can deduce the range for $x$. If $x=1$, then $y=23$. (Pair: $(1, 23)$) If $x=2$, then $y=22$. (Pair: $(2, 22)$) ... If $x=23$, then $y=1$. (Pair: $(23, 1)$) The possible values for $x$ range from 1 to 23. For each such $x$, $y$ is uniquely determined as $24-x$.
• Reasoning: Each distinct value of $x$ in the range $[1, 23]$ generates a unique ordered pair $(x, 24-x)$ where both components are positive integers. Therefore, the total number of possible outcomes (ordered pairs) is $n(S) = 23$.

Step 2: Find the Greatest Possible Product ($P_{max}$)

• What we are doing: We need to find the maximum possible value of the product $xy$ given that $x+y=24$.
• Why we are doing it: The problem condition refers to a fraction of this greatest possible product, so finding $P_{max}$ is a necessary intermediate step.
• Math: Using the AM-GM inequality for $x$ and $y$: $\frac{x+y}{2} \ge \sqrt{xy}$ Substitute $x+y=24$: $\frac{24}{2} \ge \sqrt{xy}$ $12 \ge \sqrt{xy}$ Squaring both sides (which is valid as both sides are positive): $12^2 \ge (\sqrt{xy})^2$ $144 \ge xy$ This inequality shows that the product $xy$ can be at most 144. The maximum product, $P_{max} = 144$, occurs when equality holds in the AM-GM inequality, which means $x=y$. Since $x+y=24$, this implies $x=y=12$.
• Reasoning: The AM-GM inequality provides a fundamental way to find the maximum product of two numbers when their sum is fixed. For integers, if the sum is even, the maximum occurs when the integers are equal.

Step 3: Define the Condition for Favorable Outcomes

• What we are doing: We translate the problem statement's condition for a "favorable outcome" into a mathematical inequality.
• Why we are doing it: This inequality will allow us to filter the sample space and count the outcomes that satisfy the given criteria.
• Math: The problem states that "their product is not less than $\frac{3}{4}$ times their greatest possible product". Let $P = xy$. The condition is $P \ge \frac{3}{4} P_{max}$. Substitute $P_{max} = 144$: $xy \ge \frac{3}{4} \times 144$ $xy \ge 3 \times 36$ $xy \ge 108$
• Reasoning: The phrase "not less than" mathematically translates to "greater than or equal to" ($\ge$).

Step 4: Identify the Number of Favorable Outcomes ($n(E)$)

• What we are doing: We need to find how many pairs $(x,y)$ from our sample space (where $x+y=24$ and $x,y$ are positive integers) also satisfy the condition $xy \ge 108$.
• Why we are doing it: This count represents the number of favorable outcomes needed for the probability calculation.
• Math: We have two conditions:
  1. $x+y=24 \implies y=24-x$
  2. $xy \ge 108$ Substitute $y=24-x$ from the first condition into the second: $x(24-x) \ge 108$ Expand and rearrange the terms to form a quadratic inequality: $24x - x^2 \ge 108$ $0 \ge x^2 - 24x + 108$ $x^2 - 24x + 108 \le 0$ To solve this, we find the roots of the quadratic equation $x^2 - 24x + 108 = 0$. We look for two numbers that multiply to 108 and add to -24. These numbers are -6 and -18. So, the quadratic factors as: $(x-6)(x-18) \le 0$ For the product of two terms to be less than or equal to zero, $x$ must lie between or be equal to the roots. Thus, the solution to the inequality is: $6 \le x \le 18$ Now, we count the number of positive integer values for $x$ in this range. These values are $6, 7, 8, \ldots, 18$. The number of favorable outcomes, $n(E) = (18 - 6) + 1 = 13$.
• Reasoning: Each integer $x$ in the range $[6, 18]$ corresponds to a unique positive integer $y=24-x$ such that their sum is 24 and their product is at least 108. For example, if $x=6$, $y=18$, $xy=108$. If $x=18$, $y=6$, $xy=108$. If $x=12$, $y=12$, $xy=144$. All these are valid favorable outcomes.

Step 5: Calculate the Probability ($P(E)$)

• What we are doing: We use the counts for favorable outcomes and total outcomes to calculate the probability.
• Why we are doing it: This step yields the fraction $\frac{m}{n}$ as required by the problem.
• Math: We have $n(E) = 13$ and $n(S) = 23$. $P(E) = \frac{n(E)}{n(S)} = \frac{13}{23}$ The problem states that the probability is $\frac{m}{n}$ where $\operatorname{gcd}(m, n)=1$. Here, $m=13$ and $n=23$. Since both 13 and 23 are prime numbers, their greatest common divisor is 1. Thus, the fraction $\frac{13}{23}$ is in its simplest form.

Step 6: Calculate $n-m$

• What we are doing: We perform the final arithmetic calculation requested by the question.
• Why we are doing it: This is the ultimate objective of the problem.
• Math: Using the values $m=13$ and $n=23$: $n-m = 23 - 13 = 10$

3. Common Mistakes & Tips

• Positive Integers: Always remember that "positive integers" means $x, y \ge 1$. If the problem specified "non-negative integers", the sample space would include $(0, 24)$ and $(24, 0)$, increasing $n(S)$ to 25.
• Counting Integers in a Range: When counting integers from $a$ to $b$ inclusive, the formula is $b-a+1$. A common mistake is to simply subtract $b-a$.
• Quadratic Inequality Direction: For $(x-a)(x-b) \le 0$ (with $a \le b$), the solution is $a \le x \le b$. For $(x-a)(x-b) \ge 0$, the solution is $x \le a$ or $x \ge b$. Pay close attention to the inequality sign.

4. Summary

This problem required a systematic approach combining probability fundamentals with number theory and optimization. We first established the total number of possible pairs of positive integers summing to 24. Then, we utilized the AM-GM inequality to determine the greatest possible product. This maximum product was used to formulate a quadratic inequality, which, upon solving, yielded the range of $x$ values corresponding to favorable outcomes. Counting the integers in this range gave us the number of favorable outcomes. Finally, we calculated the probability and performed the requested subtraction $n-m$.

5. Final Answer

The final answer is $\boxed{10}$, which corresponds to option (A).