https://file.aiquickdraw.com/mathbot/file/1726812890311r7z32571.jpg

Let's analyze and solve the problems from the assignment image you've uploaded.

### 1. Word Formation with "SERIES"
- **(i) Words starting with "R"**: 
  The word "SERIES" has 6 letters, with 'S', 'E', and 'R' repeating. 
  - Fixing 'R' as the first letter, we have 5 remaining letters: S, E, I, E, S.
  - To arrange these, the number of distinct permutations is given by:
    \[
    \text{Permutations} = \frac{5!}{2!2!} = \frac{120}{4} = 30
    \]

- **(ii) Words starting with "S"**:
  - Fix 'S' as the first letter, with 5 remaining: E, R, I, E, S.
  - The number of distinct permutations is:
    \[
    \frac{5!}{2!} = \frac{120}{2} = 60
    \]

- **Alternate vowels and consonants**:
  Vowels: E, I, E  
  Consonants: S, R, S  
  - Arrange consonants in the 1st, 3rd, and 5th positions: \( \frac{3!}{2!} = 3 \)
  - Arrange vowels in the 2nd, 4th, and 6th positions: \( \frac{3!}{2!} = 3 \)
  - Total number of such arrangements: \( 3 \times 3 = 9 \)

### 2. Prize Distribution
- **(i) One student gets only one prize**:
  - There are 6 prizes and 4 students. Each student can get one prize:
    \[
    \text{Ways} = P(6, 4) = \frac{6!}{(6-4)!} = \frac{6 \times 5 \times 4 \times 3}{1} = 360
    \]

- **(ii) One student can get all six prizes**:
  - Each prize can go to any of the 4 students:
    \[
    \text{Ways} = 4^6 = 4096
    \]

### 3. Find \(n\) if \(\frac{n_2}{n_3} = 3:2\)
Given this is related to combinations or binomials, please clarify the context or let me know if you'd like an assumption-based solution.

### 4. Find \(x\) if \(\frac{19}{x-2} = \frac{19}{3x+1}\)
- Cross-multiply and solve:
  \[
  19(3x+1) = 19(x-2)
  \]
  \[
  3x + 1 = x - 2
  \]
  \[
  3x - x = -2 - 1
  \]
  \[
  2x = -3 \quad \Rightarrow \quad x = -\frac{3}{2}
  \]

### 5. Find \(n\) and \(r\) if \(\binom{n}{r} = 240\) and \(\binom{n}{r+1} = 120\)
This involves solving a system of equations with binomial coefficients.

### 6. Evaluate the limit \( \lim_{x \to 0} \frac{\sqrt{1+x} - \sqrt{1-x}}{x} \)
Use L'Hopital's Rule or Taylor expansion to solve this.

### 7-10. Derivatives
- **(7)**: \( y = (\log x)^x \)
- **(8)**: \( y = \log \left( \sqrt{x+1} - \sqrt{x-1} \right) \)
- **(9)**: Parametric derivatives, \( x = a(t - \sin t), y = a(1 - \cos t) \)
- **(10)**: \( xy = e^{x+4} \)

Would you like detailed solutions to any specific problem? Let me know!

Here are some related questions you can explore:
1. What is the significance of arranging vowels and consonants alternately in permutations?
2. How are binomial coefficients used to solve combination problems?
3. How does cross-multiplication work in rational equations like in Question 4?
4. What is L'Hopital's Rule and when is it applied?
5. How do you derive parametric equations?

**Tip**: In combinatorics, always account for repeated elements when calculating permutations.

Solve the math problems provided in the image related to permutations, prize distribution, solving equations, limits, and derivatives.

This assignment covers a variety of mathematical topics including permutations, prize distribution using binomial coefficients, solving algebraic equations, evaluating limits, and finding derivatives. Detailed solutions include cross-multiplication, L'Hopital's Rule, and parametric derivatives.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation