Examples

Multinomial: Soft drinks preferred

Folks coming to a party prefer Coke (55%), Pepsi (25%), or Dew (20%). If 20 people order drinks in sequence, what is the probability that exactly 12 have Coke and 5 have Pepsi and 3 have Dew?

Solution

The multinomial coefficient ${\displaystyle \left(\genfrac{}{}{0px}{}{20}{\mathrm{12,5,3}}\right)}$ gives the number of ways to assign 20 people into bins according to preferences matching the given numbers, $C=12$ and $P=5$ and $D=3$.

Each such assignment is one sequence of outcomes. All such sequences have probability ${(0.55)}^{12}\cdot {(0.25)}^5\cdot {(0.2)}^3$.

The answer is therefore:

$$\left(\genfrac{}{}{0px}{}{20}{\mathrm{12,5,3}}\right)\cdot {(0.55)}^{12}\cdot {(0.25)}^5\cdot {(0.2)}^3\gg \gg \frac{20!}{12!5!3!}\cdot {(0.55)}^{12}\cdot {(0.25)}^5\cdot {(0.2)}^3$$