A bowl contains 10 jellybeans (four red, one blue and five white). If you pick three jellybeans from the bowl at random and without replacement, what is the probability that exactly two will be red? Express your answer as a common fraction.
First, we consider how many total sets of three jellybeans we can select, which is very simply ${10 \choose 3} = 120$, if we treat all 10 jellybeans as distinct. Now, if we have exactly 2 red jellybeans, there are ${4 \choose 2} = 6$ pairs of red jellybeans, and $5+1 = 6$ choices for the third non-red jellybean. So, there are $6 \cdot 6 = 36$ successful outcomes. So our probability is $\frac{6 \cdot 6}{120} = \frac{6}{20} = \boxed{\frac{3}{10}}$.
\frac{3}{10}
Level 5
Counting & Probability
Problem: A bowl contains 10 jellybeans (four red, one blue and five white). If you pick three jellybeans from the bowl at random and without replacement, what is the probability that exactly two will be red? Express your answer as a common fraction.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the probability of picking exactly two red jellybeans when drawing three from a bowl containing four red, one blue, and five white jellybeans, we first determine the total number of jellybeans, which is 10. 

Next, we calculate the total number of ways to choose 3 jellybeans from 10. This is given by the combination formula C(n, k), where n is the total number of items, and k is the number of items to choose. Thus, the total combinations are C(10, 3).

Now, we need to find the number of favorable outcomes where exactly two jellybeans are red. To achieve this, we must select 2 red jellybeans from the 4 available and 1 non-red jellybean from the remaining 6 (1 blue and 5 white). 

The number of ways to choose 2 red jellybeans from 4 is C(4, 2). The number of ways to choose 1 non-red jellybean from 6 is C(6, 1). 

The total number of favorable outcomes is the product of these two combinations: C(4, 2) * C(6, 1).

Now we compute these values:
C(10, 3) = 120,
C(4, 2) = 6,
C(6, 1) = 6.

Thus, the number of favorable outcomes is 6 * 6 = 36.

Finally, the probability of picking exactly two red jellybeans is the ratio of the number of favorable outcomes to the total outcomes: P(exactly 2 red) = 36 / 120.

Simplifying this fraction gives us 3 / 10. 

Therefore, the final answer is $\frac{3}{10}$.<|END_OF_LATENT|> The final answer is $\frac{3}{10}$.