Question

A police department in a small city consists of 10 officers. If the department policy is to have 5 of the officers patrolling the streets, 2 of the officers working full time at the station, and 3 of the officers on reserve at the station. How many different divisions of the 10 officers into the 3 groups are possible?

Answer 1

2,520

Explanation:

The number of possible arrangements of officers patrolling the streets is the combination of choosing 5 officers out of 10 (₁₀C₅). After choosing the patrolling officers, the number of arrangements for the officers working full time at the station is given by the combination of choosing 2 officers out of the 5 remaining (₅C₂). The remaining officers should be on reserve at the station (₃C₃). The total number of arrangements is:

`N = _(10)C_5*_5C_2*_3C_3\\N=(10!)/((10-5)!5!) *(5!)/((5-2)!2!)*(3!)/((3-3)!3!) \\N=2,520`

There are 2,520 possible divisions.