Question

a city council consists of eight democrats and eight republicans. if a committee of four people is selected, find the probability of selecting two democrats and two republicans. (type a fraction. simplify your answer.)

Explanation:

Step1: Calculate the total number of ways to select 4 people from 16

The formula for combinations is \(C(n,r)=\frac{n!}{r!(n - r)!}\), where \(n = 16\) (total number of council members: \(8+8\)) and \(r = 4\).

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Step2: Calculate the number of ways to select 2 Democrats from 8 and 2 Republicans from 8

The number of ways to select 2 Democrats from 8 is \(C(8,2)=\frac{8!}{2!(8 - 2)!}=\frac{8\times7\times6!}{2!\times6!}=\frac{8\times7}{2\times1} = 28\).
The number of ways to select 2 Republicans from 8 is also \(C(8,2)=28\).
By the multiplication principle, the number of ways to select 2 Democrats and 2 Republicans is \(C(8,2)\times C(8,2)=28\times28 = 784\).

Step3: Calculate the probability

The probability \(P\) is the number of favorable outcomes divided by the number of total outcomes. So \(P=\frac{784}{1820}\).
Simplify the fraction: \(\frac{784\div28}{1820\div28}=\frac{28}{65}\)

Answer:

\(\frac{28}{65}\)