A city council consists of six Democrats and six 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.) C
A city council consists of six Democrats and six 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.) C
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Problem Statement:**
A city council consists of six Democrats and six Republicans. If a committee of four people is selected, find the probability of selecting two Democrats and two Republicans.
---
**Solution:**
To find the probability, follow these steps:
1. **Total number of ways to choose a committee of four out of twelve:**
\[
\text{Total ways} = \binom{12}{4}
\]
2. **Ways to choose two Democrats out of six:**
\[
\text{Ways to choose Democrats} = \binom{6}{2}
\]
3. **Ways to choose two Republicans out of six:**
\[
\text{Ways to choose Republicans} = \binom{6}{2}
\]
4. **Apply the multiplication principle for probability:**
\[
\text{Favorable outcomes} = \binom{6}{2} \times \binom{6}{2}
\]
5. **Calculate the probability:**
\[
\text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total ways}}
\]
---
**Note**: Remember to simplify your final answer to a fraction.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1a4d4bf4-ffa5-40bf-a4b6-a19d4afcfff5%2Fb48b66f0-74af-44f4-84a2-8cd377dcd2e9%2F33uhgqb_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
A city council consists of six Democrats and six Republicans. If a committee of four people is selected, find the probability of selecting two Democrats and two Republicans.
---
**Solution:**
To find the probability, follow these steps:
1. **Total number of ways to choose a committee of four out of twelve:**
\[
\text{Total ways} = \binom{12}{4}
\]
2. **Ways to choose two Democrats out of six:**
\[
\text{Ways to choose Democrats} = \binom{6}{2}
\]
3. **Ways to choose two Republicans out of six:**
\[
\text{Ways to choose Republicans} = \binom{6}{2}
\]
4. **Apply the multiplication principle for probability:**
\[
\text{Favorable outcomes} = \binom{6}{2} \times \binom{6}{2}
\]
5. **Calculate the probability:**
\[
\text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total ways}}
\]
---
**Note**: Remember to simplify your final answer to a fraction.
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