There are 4 mathematicians and 7 computer scientists. We need to create a team of experts that has 5 members. At least 2 members should be mathematicians. What is the number of ways we can choose? The solution is 301. Can someone explain why?
My attempt:
4 mathematicians (Ms for short) and 7 computer scientists (CSs for short) can be members. There are $11!/5!(11−5)!=462$ combinations for assigning membership. We need to subtract those combinations with 0 or 1 M. For 0 Ms it's easy: there is only one combination with 0 Ms i.e. the one with 5 CSs. As for those with 1 M: since there are four Ms there are at least 4 different combinations. In other 4 places (one place is taken by M) we need to distribute 7 CSs. $7!/(7−4)!=7!/3!=840$ permutations there are for that. $840$ permutations of CSs for each distinct M is $3360$. There are 4 places for CSs, so to bring $3360$ from permutations to combinations we need to divide by $4!$. $3360/24=140$. Then I concluded that there are $462−1−140=321$ combinations that contain at least 2 Ms. As you can see, I am very close ($321−301$ is only $20$). So... can you detect where am I wrong?
There are $4 + 7 = 11$ people available to serve on the team. Five of them can be selected in $\binom{11}{5}$ ways. From these, we must subtract those cases in which there are fewer than two mathematicians. Since there are four mathematicians and seven computer scientists, the number of ways of selecting exactly $k$ mathematicians and $5 - k$ computer scientists is $$\binom{4}{k}\binom{7}{5 - k}$$ The number of ways of selecting no mathematicians is thus $$\binom{4}{0}\binom{7}{5}$$ and the number of ways of selecting exactly one mathematician is $$\binom{4}{1}\binom{7}{4}$$ Hence, the number of teams with five members that contain at least two mathematicians is $$\binom{11}{5} - \binom{4}{0}\binom{7}{5} - \binom{4}{1}\binom{7}{4} = 462 - 21 - 140 = 301$$ As you can see, you made a mistake when you concluded that only one team could have no mathematicians. You failed to account for the fact that we had to select which five of the seven computer scientists would serve on the team.