Probability of picking 3 different colors from 56 Skittles Suppose that you have a bag full of Skittles. In the bag are exactly 12 red Skittles, 16 orange Skittles, 10 yellow Skittles, and 18 green Skittles.  If you reach into the bag and select three Skittles at random, what's the probability that the three Skittles will be of three different colors?
If I picked a red skittle the first time, the probability would be (12/56)(44/55)(???/54) How can I account for the fact that I can choose 1 of 3 colors for the second skittle? That would change the number of skittles I can choose for the third skittle.
 A: I apologize for my previous answer...I didn't read the problem carefully.
Let's focus on the number of ways we can pick 3 skittles such that they are all the different. Let's first look at the number of ways to pick a red, orange, and yellow skittle. There are $12$ red skittles, $16$ orange skittles, and $10$ yellow skittles so there are $12 \cdot 16 \cdot 10$ different ways of picking a red, orange, and yellow skittle.
To get the total number of ways to pick 3 skittles of different colors, just consider all the ways to pick $3$ colors out of $4$ and find the number of ways for each color combination. Doing this, we get a total of
$$12 \cdot 16 \cdot 10 + 12\cdot 16 \cdot 18 + 16 \cdot 10 \cdot 18+12\cdot 10 \cdot 18$$
ways to pick $3$ skittles all of different colors.
Since there are ${12+16+10+18 \choose 3}$ ways to pick $3$ skittles in general, the probability is,
$$\frac{12 \cdot 16 \cdot 10 + 12\cdot 16 \cdot 18 + 16 \cdot 10 \cdot 18+12\cdot 10 \cdot 18}{{12+16+10+18 \choose 3}}$$
A: Skittles usually have a little s on them. We rub them out here and write the numbers 1–56 in their place so they are all distinct. The number of ways to choose 3 Skittles from 56 distinct ones is $\binom{56}3=27720$.
The number of such selections with the three colours distinct can be found case-by-case. Suppose the colours are red, orange and yellow; there are 12, 16 and 10 Skittles of the respective colours, and the number of selections with one of each is $12×16×10=1920$. Similar products apply for the other chromatically distinct combinations:


*

*$12×16×18=3456$ for red/orange/green

*$12×10×18=2160$ for red/yellow/green

*$16×10×18=2880$ for orange/yellow/green


The total number of possibilities for three differently coloured Skittles is 10416. The final probability is then
$$\frac{10416}{27720}=\frac{62}{165}=0.376\text{ (3 dp)}$$
