Question

A town is forming a committee to investigate ways to improve public safety in the town. The committee will consist of three representatives from the seven​-member town​ council, two members of a six​-person citizens advisory​ board, and three of the 10 police officers on the force. How many ways can that committee be​ formed?

Answer 1

63,000

Explanation:

Three representatives from the seven​-member town​ council can be chosen in

`C^7_3=(7!)/(3!(7-3)!)=(7!)/(3!\cdot 4!)=(4!\cdot 5\cdot 6\cdot 7)/(2\cdot 3\cdot 4!)=5\cdot 7=35`

different ways.

Two members of a six​-person citizens advisory​ board can be chosen in

`C^6_2=(6!)/(2!(6-2)!)=(6!)/(2!\cdot 4!)=(4!\cdot 5\cdot 6)/(2\cdot 4!)=5\cdot 3=15`

different ways.

Three of the 10 police officers on the force can be chosen in

`C^(10)_3=(10!)/(3!(10-3)!)=(10!)/(3!\cdot 7!)=(7!\cdot 8\cdot 9\cdot 10)/(2\cdot 3\cdot 7!)=4\cdot 3\cdot 10=120`

different ways.

In total,

`35\cdot 15\cdot 120=63,000`

different ways to form the committee