What is the probability that some people are chosen in different groups? Let's say that I have $n$ people and $m \le n$ of this people are malicious. Then I run a process that chooses 6 people randomly and assigns each of them to one of two different groups $A$ and $B$ such that each group in the end has 3 people. What is the probability that there will be at least a malicious person in every group?

You can also think of it like this: $n$ persons are split up in group $A$ containing $3$ persons, group $B$ containing $3$ persons and group $C$ containing $n-6$ persons. Now $m$ of the $n$ persons are chosen randomly and are labeled as "malicious". What is the probability that the groups $A$ and $B$ contain at least one labeled person? In that way of thinking my answer is best understood.
Let $A$ denote the event that a malicious person is in group $A$ and let $B$ denote the event that a malicious person is in group $B$. Then:$$P(A\cap B)=1-P(A^{\complement}\cup B^{\complement})=1-P(A^{\complement})-P(B^{\complement})+P(A^{\complement}\cap B^{\complement})=$$$$1-\frac{\binom30\binom{n-3}{m}}{\binom{n}{m}}-\frac{\binom30\binom{n-3}{m}}{\binom{n}{m}}+\frac{\binom60\binom{n-6}{m}}{\binom{n}{m}}=$$$$\frac{\binom{n}{m}-2\binom{n-3}{m}+\binom{n-6}{m}}{\binom{n}{m}}$$
