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There are 5 blue, 7 green, and some yellow bottles in a sack. If one bottle is drawn at random, the probability of it being yellow is \(\frac{1}{3}\). Three bottles are taken out at random. What is the probability that all three are of different colours?

1. 37/136
2. 35/136
3. 39/143
4. 35/143

Correct answer: 35/136

Solution

Let the number of yellow bottles be \(y\). Since \(\frac{y}{5+7+y}=\frac{1}{3}\), we get \(3y=12+y\Rightarrow y=6\). For all three bottles to be different, we need one blue, one green, and one yellow: \(5\times 7\times 6=210\) ways. Total ways to choose 3 bottles from 18 is \(\binom{18}{3}=816\), so probability is \(\frac{210}{816}=\frac{35}{136}\).

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