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 :

Question

A bag contains 10 balls out of which $ k $ are red and $ (10-k) $ are black, where $ 0 \le k \le 10 $. If three balls are drawn at random without replacement and all of them are found to be black, then the probability that the bag contains 1 red and 9 black balls is:

Answer 1

Given two positive integers summing to 24, we determine the probability that their product is at least $ \frac{3}{4} $ of their maximum possible product. We then find the difference $ n - m $, where the probability is expressed as the simplified fraction $ \frac{m}{n} $.

Step 1: Let the two integers be $ x $ and $ y $. We are given $ x + y = 24 $, so $ y = 24 - x $. The product $ P $ is $ P = x(24 - x) = 24x - x^2 $.

Step 2: The product $ P $ is a quadratic function. The maximum product occurs at the vertex, where $ x = -\frac{b}{2a} = \frac{24}{2} = 12 $. When $ x = 12 $, $ y = 24 - 12 = 12 $. The maximum product is $ P_{\text{max}} = 12 \times 12 = 144 $.

Step 3: The condition is $ x(24 - x) \geq \frac{3}{4} \times 144 = 108 $. This simplifies to $ 24x - x^2 \geq 108 $, or $ x^2 - 24x + 108 \leq 0 $.

Step 4: Solve the quadratic equation $ x^2 - 24x + 108 = 0 $. Using the quadratic formula, $ x = \frac{24 \pm \sqrt{24^2 - 4 \times 1 \times 108}}{2} = \frac{24 \pm \sqrt{576 - 432}}{2} = \frac{24 \pm \sqrt{144}}{2} = \frac{24 \pm 12}{2} $. The roots are $ x = 18 $ and $ x = 6 $.

Step 5: The inequality $ x^2 - 24x + 108 \leq 0 $ holds for $ 6 \leq x \leq 18 $.

Step 6: We count the integer solutions for $ x $ in the range [6, 18]. The pairs are $ (6,18), (7,17), \ldots, (18,6) $. The number of integer solutions for $ x $ is $ 18 - 6 + 1 = 13 $.

Step 7: The total number of pairs of positive integers $ (x, y) $ such that $ x + y = 24 $ is 23 (when $ x $ ranges from 1 to 23). The probability that their product is not less than $ \frac{3}{4} $ times their greatest product is $ \frac{13}{23} $. Thus, $ m = 13 $ and $ n = 23 $. The difference is $ n - m = 23 - 13 = 10 $.

The final answer is 10.

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 :

The Correct Option is D

Solution and Explanation

Top Questions on Probability

Questions Asked in JEE Main exam