Hair color probability 50% of the population have the hair colour gold, 40% have brown hair and 10% have black hair.
Select 10 people randomly on the street.  What is the probability that there are 5 people who have gold hair and 4 people who have have brown hair and 1 person who has black hair?
My guess is $ \binom{10}{5}(0.5)^5 \binom{5}{4}(0.4)^4 \binom{1}{1}(0.1)^1=0.1008$, however I don't know if it is correct, what I know so far is that the order of selection and the combinations should be considered, like $(0.5)^5(0.4)^4(0.1)^1$ is obviously wrong, because we may, but how to deal with probability and combinations at the same time?
 A: It is correct. First, there are $\binom{10}{5}$ ways that you may get 5 people with golden hair from 10. Now there are $10-5=5$ left, and there are $\binom{5}{4}$ ways to have four people with brown hair in the remaining 5. Then you have 1 left with only 1 choice. Finally you multiply the probabilities for each case.
$$\binom{10}{5}(0.5)^5 \binom{5}{4}(0.4)^4 \binom{1}{1}(0.1)^1=\frac{10!}{5!~4!~1!}(0.5)^5(0.4)^4(0.1)^1=0.1008$$
A: If you are confused by combinatoric problems, it is a good idea to pick a simpler problem, say there are only two hair colours, gold and brown, then there is $2/3$ probability of gold, and we ask for three people, then you may have
$$
\begin{align}
p(GGG)&=(2/3)^3\\
p(GGB)&=(2/3)^2\times 1/3\\
p(GBG)&=2/3\times 1/3\times 2/3\\
p(GBB)&=2/3\times (1/3)^2\\
p(BGG)&=1/3\times(2/3)^2\\
p(BGB)&=1/3\times 2/3\times 1/3\\
p(BBG)&=(1/3)^3\times 2/3\\
p(BBB)&=(1/3)^3\\
\end{align}
$$
so the probability of GGB, say, in any order, is $3\times(2/3)^2\times 1/3$. It's difficult to build intuition if the numbers are big and there are extra complicating factors, so it's better to do easy problems first.
