A bag contains 5 red, 4 blue, and 3 green balls. Three balls are drawn at random. What is the probability that all three balls are the same color?

The combination formula is $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.

$$\binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10$$ $$\binom{4}{3} = \frac{4 \times 3 \times 2}{3 \times 2 \times 1} = 4$$ $$\binom{3}{3} = \frac{3 \times 2 \times 1}{3 \times 2 \times 1} = 1$$ $$\binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220$$

$$P(\text{all three same colour}) = \frac{\binom{5}{3} + \binom{4}{3} + \binom{3}{3}}{\binom{12}{3}}$$ $$P(\text{all three same colour}) = \frac{10 + 4 + 1}{220}$$

$$P(\text{all three same colour}) = \frac{15}{220}$$ $$P(\text{all three same colour}) = \frac{3}{44}$$

The probability that all three balls have the same colour is $\frac{3}{44}$.