The problem is as follows:
The dean of science wants to select a committee consisting of mathematicians and physicists to discuss a new curriculum. There are $15$ mathematicians and $20$ physicists at the faculty; how many possible committees of $8$ members are there, if there must be more mathematicians than physicists (but at least one physicist) on the committee?
The given solution splits up the choices into the cases where there is $1$ physicist, $2$ physicists, and $3$ physicists. Then it's simply a matter of summing up $\binom{15}{5}\cdot\binom{20}{3}$ and $\binom{15}{6}\cdot\binom{20}{2}$ and $\binom{15}{7}\cdot\binom{20}{1}$, giving $4503070$.
This is what I tried to do: first I pick a physicist, making sure I have at least one on the team. I have $20$ choices for this. Then I pick $5$ mathematicians, making sure there are more mathematicians than physicists on the team. I have $15\cdot14\cdot13\cdot12\cdot11$ choices for this. Then I pick two people from the $29$ people that are left because it doesn't matter whether they are mathematicians or physicists. I have $29\cdot28$ choices for this. After I multiply all those numbers, I need to divide by the number of ways I can permutate them, because order doesn't matter. I get $$\dfrac{20\cdot15\cdot14\cdot13\cdot12\cdot11\cdot29\cdot28}{8!} = 145145,$$ which is obviously not the right solution. Where did I go wrong?
Thanks in advance.
You made a calculation error. $$\frac{20 \cdot 15 \cdot 14 \cdot 13 \cdot 12 \cdot 11 \cdot 29 \cdot 28}{8!} = \color{red}{1161160}$$ Let's examine your numerator: $$20 \cdot 15 \cdot 14 \cdot 13 \cdot 12 \cdot 11 \cdot 29 \cdot 28 = 20 \cdot \frac{15!}{10!} \cdot \frac{29!}{27!} = \binom{20}{1} \cdot \binom{15}{5} \cdot 5! \cdot \binom{29}{2} \cdot 2!$$ Since the order in which we choose the five designated mathematicians does not matter, we should divide your numerator by $5!$. Since the order in which the two additional committee members does not matter, we should also divide your numerator by $2!$. Doing so yields $$\binom{20}{1}\binom{15}{5}\binom{29}{2} = 24384360 = \frac{8!}{5!2!} \cdot 1161160$$ which is too large.
Let's see why.
A committee of eight people drawn from $20$ physicists and $15$ mathematicians that contains a majority of mathematicians but at least one physicist must contain either five mathematicians and three physicists or six physicists and two physicists or seven mathematicians and one physicist. The number of ways of selecting exactly $k$ mathematicians and $8 - k$ physicists from the $15$ mathematicians and $20$ physicists is $$\binom{15}{k}\binom{20}{8 - k}$$ Hence, the number of admissible committees is $$\binom{15}{5}\binom{20}{3} + \binom{15}{6}\binom{20}{2} + \binom{15}{7}\binom{20}{1} = 450370$$ By designating a particular physicist as the physicist on the committee, you count each committee with $p$ physicists $p$ times, once for each way you could designated one of them as the designated physicist. By designating five of the mathematicians on the committee as the five designated mathematicians on the committee, you count each committee with $m$ mathematicians $\binom{m}{5}$ times, once for each way you could designate five of the $m$ mathematicians as the designated mathematicians. Observe that $$\binom{20}{1}\binom{15}{5}\binom{29}{2} = \color{red}{\binom{5}{5}\binom{3}{1}}\binom{15}{5}\binom{20}{3} + \color{red}{\binom{6}{5}\binom{2}{1}}\binom{15}{6}\binom{20}{2} + \color{red}{\binom{7}{5}\binom{1}{1}}\binom{15}{7}\binom{20}{1}$$