Multinomial Coefficients and applications 10/12

How many ways are there of assigning 10 police officers to 3 different tasks:

• Patrol, of which there must be 5.
• Office, of which there must be 2.
• Reserve, of which there will be 3.

We saw how the answer comes out to be:

$$\binom{10}{5} \times \binom{5}{2} \times \binom{3}{3} =\frac{10!}{ 5! 3! 2!}$$

In general there will $\frac{n!}{k_1!k_2! ...k_r!}$ ways divide up n objects into r different categories so that there will be k₁ objects in category 1, k₂ in category 2, and so on ( $k_1+k_2+\cdots +k_r=n$).