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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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(P_(3)^(5)*C_(2)^(6)+P_(4)^(5)*C_(2)^(4))/(P_(4)^(9))

Certainly! Below is the transcription of the given mathematical expression for an educational website, along with a detailed explanation of its components.

---

### Mathematical Expression

Given the following expression:

\[
\frac{P^5_3 \cdot C^6_2 + P^5_4 \cdot C^4_2}{P^9_4}
\]

#### Explanation:

1. **Permutations and Combinations:**
   - \( P^n_r \) represents the number of permutations of \( n \) items taken \( r \) at a time.
   - \( C^n_r \) represents the number of combinations of \( n \) items taken \( r \) at a time.

2. **Numerator:**
   - The expression in the numerator consists of two terms:
     - The first term is \( P^5_3 \cdot C^6_2 \):
       - \( P^5_3 \) denotes the permutations of 5 items taken 3 at a time.
       - \( C^6_2 \) denotes the combinations of 6 items taken 2 at a time.
     - The second term is \( P^5_4 \cdot C^4_2 \):
       - \( P^5_4 \) denotes the permutations of 5 items taken 4 at a time.
       - \( C^4_2 \) denotes the combinations of 4 items taken 2 at a time.
   - These terms are added together.

3. **Denominator:**
   - The denominator \( P^9_4 \) represents the permutations of 9 items taken 4 at a time.

#### Detailed Steps:

- **Step 1**: Calculate each component individually.
  - Find the values of \( P^5_3 \), \( C^6_2 \), \( P^5_4 \), \( C^4_2 \), and \( P^9_4 \).
  - Use the formulas:
    - \( P^n_r = \frac{n!}{(n-r)!} \)
    - \( C^n_r = \frac{n!}{r!(n-r)!} \)
    
- **Step 2**: Substitute the values back into the expression.

- **Step 3**: Perform the multiplication and addition in the numerator.

- **Step 4**: Divide the
Transcribed Image Text:Certainly! Below is the transcription of the given mathematical expression for an educational website, along with a detailed explanation of its components. --- ### Mathematical Expression Given the following expression: \[ \frac{P^5_3 \cdot C^6_2 + P^5_4 \cdot C^4_2}{P^9_4} \] #### Explanation: 1. **Permutations and Combinations:** - \( P^n_r \) represents the number of permutations of \( n \) items taken \( r \) at a time. - \( C^n_r \) represents the number of combinations of \( n \) items taken \( r \) at a time. 2. **Numerator:** - The expression in the numerator consists of two terms: - The first term is \( P^5_3 \cdot C^6_2 \): - \( P^5_3 \) denotes the permutations of 5 items taken 3 at a time. - \( C^6_2 \) denotes the combinations of 6 items taken 2 at a time. - The second term is \( P^5_4 \cdot C^4_2 \): - \( P^5_4 \) denotes the permutations of 5 items taken 4 at a time. - \( C^4_2 \) denotes the combinations of 4 items taken 2 at a time. - These terms are added together. 3. **Denominator:** - The denominator \( P^9_4 \) represents the permutations of 9 items taken 4 at a time. #### Detailed Steps: - **Step 1**: Calculate each component individually. - Find the values of \( P^5_3 \), \( C^6_2 \), \( P^5_4 \), \( C^4_2 \), and \( P^9_4 \). - Use the formulas: - \( P^n_r = \frac{n!}{(n-r)!} \) - \( C^n_r = \frac{n!}{r!(n-r)!} \) - **Step 2**: Substitute the values back into the expression. - **Step 3**: Perform the multiplication and addition in the numerator. - **Step 4**: Divide the
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