Question

Let the sum of two positive integers be 24. If the probability, that their product is not less than $\frac{3}{4}$ times their greatest positive product, is $\frac{m}{n}$, where $\gcd(m, n) = 1$, then $n - m$ equals :

• A. 9
• B. 11
• C. 8
• D. 10

Answer 1

The Correct Option is D

Given \( x + y = 24 \), \( x, y \in \mathbb{N} \), the greatest product occurs at:

\[ x = y = 12 \implies \text{Maximum Product} = 144. \]

Step 1: Define the condition:

\[ xy \geq \frac{3}{4} \cdot 144 \implies xy \geq 108. \]

Step 2: List favorable pairs:

\[ (13, 11), (12, 12), (14, 10), (15, 9), (16, 8), (17, 7), (18, 6), (6, 18), (7, 17), (8, 16), (9, 15), (10, 14), (11, 13). \]

Step 3: Total cases and favorable cases:

There are \( 13 \) favorable cases out of \( 23 \) total cases.

\[ \text{Probability} = \frac{13}{23}. \]

Step 4: Calculate:

\[ m = 13, \quad n = 23 \implies n - m = 10. \]

Final Answer:

\[ \boxed{10.} \]