A bag contains green and pink balls in the ratio 7 : 5. Two balls are drawn together at random from the bag, and the probability that both balls are pink is \(\frac{1}{6}\). Find the total number of balls in the bag.
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 36
Take the counts from the ratio as \(7k\) green and \(5k\) pink. Drawing two balls is a selection, so P(both pink) \(=\) ways to choose 2 pink ÷ ways to choose any 2.
Total balls \(=12k\)
\(\displaystyle \text{P(both pink)}=\frac{{}^{5k}C_2}{{}^{12k}C_2}=\frac{5k(5k-1)}{12k(12k-1)}\)
\(\displaystyle \frac{5(5k-1)}{12(12k-1)}=\frac{1}{6}\)
\(30(5k-1)=12(12k-1)\)
\(150k-30=144k-12\), so \(6k=18\) and \(k=3\)
Total balls \(=12\times3=36\)