3. A researcher has 14 subjects taking part in an experiment. They will randomly assign 5 subjects to the first group, 5 to the second, and 4 to the third. How many ways are there to choose which participants are assigned to each group? Leave your answer in exponential or factorial form.
3. A researcher has 14 subjects taking part in an experiment. They will randomly assign 5 subjects to the first group, 5 to the second, and 4 to the third. How many ways are there to choose which participants are assigned to each group? Leave your answer in exponential or factorial form.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
Related questions
Question
discrete math
![**Problem:**
A researcher has 14 subjects taking part in an experiment. They will randomly assign 5 subjects to the first group, 5 to the second, and 4 to the third. How many ways are there to choose which participants are assigned to each group? *Leave your answer in exponential or factorial form.*
---
**Solution Explanation:**
To determine the number of ways to assign the subjects to the groups, we can use combinations. The total number of subjects is 14, and we need to decide how to divide them into groups of 5, 5, and 4.
1. **Choose 5 subjects for the first group:**
\[
\binom{14}{5}
\]
2. **Choose 5 of the remaining 9 subjects for the second group:**
\[
\binom{9}{5}
\]
3. **The last 4 subjects automatically go to the third group:**
\[
\binom{4}{4} = 1
\]
The total number of ways to assign the subjects to these groups is the product of these combinations:
\[
\binom{14}{5} \times \binom{9}{5} \times \binom{4}{4}
\]
In factorial form, this is expressed as:
\[
\frac{14!}{5!5!4!}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7cef3050-b0aa-40af-9112-57cab04a18ba%2F87bbe8ae-b848-4385-92ab-dbd06f7d7f9d%2F4rmjtie_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem:**
A researcher has 14 subjects taking part in an experiment. They will randomly assign 5 subjects to the first group, 5 to the second, and 4 to the third. How many ways are there to choose which participants are assigned to each group? *Leave your answer in exponential or factorial form.*
---
**Solution Explanation:**
To determine the number of ways to assign the subjects to the groups, we can use combinations. The total number of subjects is 14, and we need to decide how to divide them into groups of 5, 5, and 4.
1. **Choose 5 subjects for the first group:**
\[
\binom{14}{5}
\]
2. **Choose 5 of the remaining 9 subjects for the second group:**
\[
\binom{9}{5}
\]
3. **The last 4 subjects automatically go to the third group:**
\[
\binom{4}{4} = 1
\]
The total number of ways to assign the subjects to these groups is the product of these combinations:
\[
\binom{14}{5} \times \binom{9}{5} \times \binom{4}{4}
\]
In factorial form, this is expressed as:
\[
\frac{14!}{5!5!4!}
\]
Expert Solution
Step 1
Total subjects = 14
3 groups :
1: 5 subjects
2: 5 subjects
3: 4 subjects
No. of ways = ?
Step by step
Solved in 2 steps
Recommended textbooks for you
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON