Probability
Probability
Allen Star Batch
Grade 12
Question:
A boy has a collection of the blue and green marbles. The number of blue marbles belong to the set $\{2, 3, 4, \ldots, 13\}$. If two marbles are chosen simultaneously and random from his collection, then the probability that they have different colours is $1/2$. Possible number of blue marbles is:
Step-by-Step Solution
Key Concept: The discriminant of a quadratic in one variable must be a perfect square to yield integer solutions; use this algebraic constraint on the discriminant.
Let $a$ be the number of blue marbles and $b$ be the number of green marbles. Given $\frac{ab}{^{a+b}C_2} = \frac{1}{2}$, we derive $(a+b)(a+b-1) = 4ab$, which simplifies to $b^2 - (2a+1)b + a^2 - a = 0$. For integer solutions, $b^2 - (2a+1)b + a^2 - a = (2a+1)^2 - 4(a^2-a)$ must be a perfect square, giving $a ∈ \{3, 6, 10\}$.
Correct Answer: 1,2,3