Answer to Question #348526 in Statistics and Probability for Anand

Question

Suppose there are three different types of cakes that are sold at Tina’s Bakery. There are 4 cheesecakes, 5 vanilla sponge cakes and 2 black forest cakes. Tom takes at random 2 of these cakes without replacement. What is the probability that he takes 2 different types of cakes upon his selection?

Answer 1

$$P(2\ different)=\dfrac{\dbinom{4}{1}\dbinom{5}{1}\dbinom{2}{0}}{\dbinom{4+5+2}{2}}$$ $$+\dfrac{\dbinom{4}{1}\dbinom{5}{0}\dbinom{2}{1}}{\dbinom{4+5+2}{2}}+\dfrac{\dbinom{4}{0}\dbinom{5}{1}\dbinom{2}{1}}{\dbinom{4+5+2}{2}}$$

$$=\dfrac{4(5)(1)+4(1)(2)+1(5)(2)}{55}=\dfrac{38}{55}$$