Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Problem: Evaluate the Expression**
\[
\frac{{^6C_4}}{{^6P_4}}
\]
**Options:**
- A. \( 24 \)
- B. \( 4 \)
- C. \( \frac{1}{24} \)
- D. \( \frac{1}{4} \)
**Explanation:**
- **Combinations \( ^nC_r \):** This represents the number of ways to choose \( r \) items from \( n \) items without regard to order. The formula is:
\[
^nC_r = \frac{n!}{r!(n-r)!}
\]
- **Permutations \( ^nP_r \):** This represents the number of ways to choose \( r \) items from \( n \) items with regard to order. The formula is:
\[
^nP_r = \frac{n!}{(n-r)!}
\]
In the problem, substitute \( n = 6 \) and \( r = 4 \) into the formulas for combinations and permutations to evaluate the expression.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc6cb1990-6e90-47a1-997c-36a24738006c%2Ffac10b28-5368-42a3-962f-8e0c0763df01%2Fvjvz87_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem: Evaluate the Expression**
\[
\frac{{^6C_4}}{{^6P_4}}
\]
**Options:**
- A. \( 24 \)
- B. \( 4 \)
- C. \( \frac{1}{24} \)
- D. \( \frac{1}{4} \)
**Explanation:**
- **Combinations \( ^nC_r \):** This represents the number of ways to choose \( r \) items from \( n \) items without regard to order. The formula is:
\[
^nC_r = \frac{n!}{r!(n-r)!}
\]
- **Permutations \( ^nP_r \):** This represents the number of ways to choose \( r \) items from \( n \) items with regard to order. The formula is:
\[
^nP_r = \frac{n!}{(n-r)!}
\]
In the problem, substitute \( n = 6 \) and \( r = 4 \) into the formulas for combinations and permutations to evaluate the expression.
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