Problem 24 · 2011 Math Kangaroo
Stretch
Counting & Probability
perfect-square
A box contains red and green balls. If two balls are drawn at random, the probability that they are the same colour is \(\tfrac{1}{2}\). Which of the following could be the total number of balls in the box?
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Answer: A — 81
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Hint 1 of 2
Set the chance of two same-colour equal to the chance of two different.
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Hint 2 of 2
The condition simplifies to (r − g)² = r + g, so the total must be a perfect square.
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Approach: reduce the probability condition to a perfect-square total
- P(same) = 1/2 means C(r,2)+C(g,2) = rg, which simplifies to (r−g)² = r+g = n.
- So the total n must be a perfect square.
- Among the choices only 81 = 9² qualifies, so the total could be 81.
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