Separating 18 people into 5 teams A teacher wants to divide her class of 18 students into 5 teams to work on projects, 
with two teams of 3 students each and three teams of 4 students each.
a) In how many ways can she do this, if the teams are not numbered?
b) What is the probability that two of the students, Mia and Max, will be on the same team?
[This is not a homework problem.]

Form each team one at a time.
$$\begin{align}
\text{First team: }&\binom{18}{3}\\
\text{Second team: }&\binom{15}{3}\\
\text{Third team: }&\binom{12}{4}\\
\text{Fourth team: }&\binom{8}{4}\\
\text{Fifth team: }&\binom{4}{4}
\end{align}$$
Then apply the rule of products.

The probability of any given student going into team one or two is $\frac{3}{18}$. The probability of a student going into team three, four, or five is $\frac{4}{18}$. So the probability of two students both going into team one is $\frac{3}{18}\frac{3}{18}$. Use the rule of sums to compute the total probability that they'll end up in the same team out of any of the five teams:
$$P=2\frac{3}{18}\frac{3}{18}+3\frac{4}{18}\frac{4}{18}$$
