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.