15. Kiran is designing a game which involves a bag of twenty-one marbles. Some of the marbles are blue, the rest are red. To play the game, two marbles are drawn out. The game is won if at least one red marble is drawn. To ensure the probability of the game being won is exactly one-half, Kiran uses $B$ blue marbles and $R$ red marbles. What is the value of $B^{2}+R^{2}$ ?
SOLUTION
261

The probability of selecting blue then blue is 0.5 .
Therefore, $\frac{B}{21} \times \frac{B-1}{20}=0.5$.
This leads to the quadratic equation $B^{2}-B-210=0$.
This factorises as $(B-15)(B+14)=0$ with $B=15$ as the only positive solution.
This means $B=15$ and $R=6$, so $B^{2}+R^{2}=225+36=261$.