What is the probability that Carlos's purchase of 5 CDs includes at least one rap, country, and heavy metal CD out of a total of 12? Full Problem
Carlos has chosen $12$ different CDs he would like to buy: $4$ are rap music, $5$ are country music, and $3$ are heavy metal music. (Carlos has very eclectic tastes in music!) Unfortunately, he has only enough money to afford to buy $5$ of them (they all cost the same price). So he selects $5$ of them at random. What is the probability that his purchase includes at least one CD from each of the three categories?
My Response
First, there are a total of $\dbinom{12}5$ total ways for Carlos to choose, without order, $5$ CDs from $12$ CDs. Then, there are a total of $\dbinom41 \dbinom51 \dbinom31 \dbinom92$ ways for Carlos to choose at least one CD from each category. This simplifies to $\dfrac{30}{11}$, which is obviously not correct. What went wrong in my process?
 A: By designating a particular CD from each genre as the CD from that genre, you count each genre from which Carlos selects more than one CD multiple times.
For instance, if the rap selections are A and B, the country selection is C, and the heavy metal selections are D and E, your method counts this choice $2 \cdot 1 \cdot 2 = 4$ times.
$$
\begin{array}{c c c c}
\text{rap} & \text{country} & \text{heavy metal} & \text{additional CDs}\\
\hline
A & C & D & B, E\\
A & C & E & B, D\\
B & C & D & A, E\\
B & C & E & A, D
\end{array}
$$
To avoid this problem, you can either use the Inclusion-Exclusion Principle or you can consider how many CDs of each type are selected.

*

*$3$ rap, $1$ country, $1$ heavy metal

*$2$ rap, $2$ country, $1$ heavy metal

*$2$ rap, $1$ country, $2$ heavy metal

*$1$ rap, $3$ country, $1$ heavy metal

*$1$ rap, $2$ country, $2$ heavy metal

*$1$ rap, $1$ country, $3$ heavy metal

A: From another view point:
Because we calculate a probability, each item should be seen as distinct, even if they are in same category. So, let's write their generating functions forms thinking this fact !

*

*Generating function of selecting at least one rap music : $$\binom{4}{1}x^1+\binom{4}{2}x^2+\binom{4}{3}x^3+\binom{4}{4}x^4=(1+x)^4 -1$$


*Generating function of selecting at least one country music : $$\binom{5}{1}x^1+\binom{5}{2}x^2+\binom{5}{3}x^3+\binom{5}{4}x^4+\binom{5}{5}x^5=(1+x)^5 -1$$


*Generating function of selecting at least one heavy metal music : $$\binom{3}{1}x^1+\binom{3}{2}x^2+\binom{3}{3}x^3=(1+x)^3 -1$$
Now , find the coefficient of $x^5$ in the expansion of $$[(1+x)^4-1][(1+x)^5-1][(1+x)^3-1]$$
Calculation:
$$\frac{[x^5]([(1+x)^4-1][(1+x)^5-1][(1+x)^3-1])}{\binom{12}{5}}=\frac{590}{792}=0,7449...$$
