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JEE Main 20248 AprilMorning Shift
Exam
JEE Main
Year
2024
Date
8 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Probability

JEE Main 2024 · 8 April · Morning Shift · Mathematics · Probability

Let the sum of two positive integers be 24 . If the probability,...

Let the sum of two positive integers be 24 . If the probability, that their product is not less than 34\frac{3}{4} times their greatest possible product, is mn\frac{m}{n}, where gcd⁡(m,n)=1\operatorname{gcd}(m, n)=1, then n−mn-m equals

Options

  1. A10
  2. B9
  3. C11
  4. D8

Correct answer

A. 10

Step-by-step solution

x+y=24,x,y∈NAM>GM⇒xy≤144xy≥108\begin{aligned} & x+y=24, x, y \in N \\ & A M>G M \Rightarrow x y \leq 144 \\ & x y \geq 108\end{aligned} Favorable pairs of (x,y)(x, y) are (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)\begin{aligned} & (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)\end{aligned} i.e. 13 casesTotal choices for x+y=24\mathrm{x}+\mathrm{y}=24 is 23  Probability =1323=mnn−m=10\begin{aligned} & \text{ Probability }=\frac{13}{23}=\frac{m}{n} \\ & n-m=10\end{aligned}
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