The question is :
Suppose that a school band contains $10$ students from the freshman class, $20$ students from the sophomore class, $30$ students from the junior class, and $40$ students from the senior class. If $15$ students are selected at random from the band, what is the probability that at least one students from each of the four classes?
I think the total number of outcomes is: ${100 \choose 15}$
I suppose that $4$ students are selected from each of the four classes. Now I have $96$ students taken $11$, so it is: $96\choose 11$
The probability is: $13/37345$
I am so confused. Please tell me if I was right or wrong and give me a hint!
An alternative to inclusion-exclusion is to use the generating function $$\left(\sum_{k=1}^{10} \binom{10}{k} x^k\right)\left(\sum_{k=1}^{20} \binom{20}{k} x^k\right)\left(\sum_{k=1}^{30} \binom{30}{k} x^k\right)\left(\sum_{k=1}^{40} \binom{40}{k} x^k\right)=\left((1+x)^{10}-1\right)\left((1+x)^{20}-1\right)\left((1+x)^{30}-1\right)\left((1+x)^{40}-1\right).$$ The coefficient of $x^{15}$ turns out to be $200911429227922000$, which yields probability $$\frac{200911429227922000}{\binom{100}{15}} = \frac{14165531225}{17861970522}$$