I have a drawer with 6 forks, 6 spoons, and 6 knives in it. If I reach in and randomly remove three pieces of silverware, what is the probability that I get one fork, one spoon, and one knife?
First, we can find the denominator of our fraction.  There are a total of $\dbinom{18}{3}=816$ ways to choose 3 pieces of silverware out of 18.  To find the numerator, we need to count the number of ways to choose one piece of each type of silverware.  There are 6 ways we could choose a fork, 6 ways to choose a spoon, and 6 ways to choose a knife, making a total of $6\cdot 6 \cdot 6=216$ ways to choose a fork, a knife, and a spoon so our final probability is $\frac{216}{816}=\boxed{\frac{9}{34}}$.
\frac{9}{34}
Level 4
Counting & Probability
Problem: I have a drawer with 6 forks, 6 spoons, and 6 knives in it. If I reach in and randomly remove three pieces of silverware, what is the probability that I get one fork, one spoon, and one knife?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the probability of drawing one fork, one spoon, and one knife from the drawer, we first determine the total number of ways to choose any three pieces of silverware from the 18 available (6 forks + 6 spoons + 6 knives). The total combinations can be calculated using the combination formula C(n, k), where n is the total items and k is the items to choose. 

C(18, 3) = 18! / (3!(18-3)!) = 18! / (3! * 15!) = (18 * 17 * 16) / (3 * 2 * 1) = 816.

Next, we calculate the number of favorable outcomes for drawing one fork, one spoon, and one knife. The number of ways to choose one fork from six is C(6, 1), the same applies for spoons and knives.

C(6, 1) = 6 (for forks),
C(6, 1) = 6 (for spoons),
C(6, 1) = 6 (for knives).

The total number of favorable outcomes is the product of these combinations:

6 (forks) * 6 (spoons) * 6 (knives) = 216.

Now, we can find the probability by dividing the number of favorable outcomes by the total outcomes:

Probability = Favorable outcomes / Total outcomes = 216 / 816.

To simplify this fraction, we find the greatest common divisor (GCD) of 216 and 816. The GCD is 24.

Dividing both the numerator and the denominator by 24 gives:

216 / 24 = 9,
816 / 24 = 34.

Thus, the simplified probability is:

Probability = 9 / 34.

This leads us to the final answer of the probability of drawing one fork, one spoon, and one knife, which is $\frac{9}{34}$.<|END_OF_LATENT|> The final answer is $\frac{9}{34}$.