A bag contains 5 red balls, 4 green balls, and 3 blue balls. If two balls are drawn at random simultaneously without replacement, the probability that both balls are red is expressed in simplest fraction form as $\frac{p}{q}$. The value of $(p + q)$ is ________ (answer in integer).
Answer Key:
38
Step-by-Step Solution & Explanation
1. Total number of balls $= 5 + 4 + 3 = 12$.
2. Total possible outcomes when drawing 2 balls:
$n(S) = \binom{12}{2} = \frac{12 \times 11}{2} = 66$
3. Favourable outcomes of drawing 2 red balls from 5 red balls:
$n(E) = \binom{5}{2} = \frac{5 \times 4}{2} = 10$
4. Probability $P = \frac{10}{66} = \frac{5}{33}$.
Here $p = 5$ and $q = 33$, so $p + q = 5 + 33 = 38$.