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Democrats and Republicans on Task Force

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A certain city has experienced a relatively rapid increase in traffic congestion in recent times. The mayor has decided that it's time to do something about the problem and decides to form a task force to research the causes of the problem and come up with suggestions for dealing with it. The mayor first compiles a list of names of 21 residents who will be considered for membership on the task force. Twelve of those 21 people are Republicans, and the other 9 are Democrats. How many ways are there to form a task force of 7 people from that list of 21 candidates if at least one member of the task force has to be a Republican and at least one member has to be a Democrat?

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Solution Preview

The total number of ways to form a task force of 7 members from a group of 21 people is C(21, 7), the number of combinations of 7 "objects" taken from a set of 21 "objects" (sometimes called "21 choose 7" or denoted by 21C7).
Recall that:

C(21, 7) = (21!)/[(7!)(21 - 7)!]

= (21!)/[(7!)(14!)]

Now 21! (21 factorial), which is the product of the first 21 positive integers,
can be written as (21)(20)(19)(18)(17)(16)(15)(14!). Thus

C(21, 7) = (21)(20)(19)(18)(17)(16)(15)(14!)/[(7!)(14!)]

Canceling the 14! in the numerator against the 14! in the denominator, we find that

C(21, 7) = (21)(20)(19)(18)(17)(16)(15)/(7!)

= (21)(20)(19)(18)(17)(16)(15)/[(7)(6)(5)(4)(3)(2)(1)]

Note that 21 = 7(3), so dividing the numerator and denominator
by 21 (= 7(3)), we have

C(21, 7) = ...

Solution Summary

First, the general problem of forming a task force of 7 people from a group of 21 people is addressed. Then the requirement that there be at least one Democrat and at least one Republican on the take force is dealt with.

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