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

Answer: (D)

10

Answer 2

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:

$\boxed{10.}$