General And Middle Terms

Coefficient of constant term is

The correct answer is (B): .
Now ,

Coefficient of

Step 1: General term: Tᵣ+₁ = ¹⁵Cᵣ(x⁴)¹⁵⁻ʳ(-(1)/(x³))ʳ
Step 2: Power of x: x⁶⁰⁻⁷ʳ
Step 3: 60-7r=32 r=4
Step 4: Coefficient: ¹⁵C₄(-1)⁴ = -15C₂

Step 1: Ratio of terms. T₇Tₙ-₆ = (2¹/³3⁻¹/³)¹³⁻ⁿ
Step 2: Simplification. 2¹/³3⁻¹/³ = (6)¹/³ (6)⁽¹³⁻ⁿ⁾/³ = (1)/(6)
Step 3: Solving. (13-n)/(3) = -1 n = 9

Step 1: Use the binomial expansion formula.
The expansion of is:
Step 2: Find the middle term.
The middle term occurs when . Thus, the middle term is:
Step 3: Conclusion.
The coefficient of the middle term is 216. Final Answer:

Step 1: Understanding the Concept:
First, we find the value of by equating the general terms of the binomial expansion. Then, we use the general term formula for a negative binomial expansion to find the required coefficient.

Step 2: Key Formula or Approach:
1. General term for is .
2. General term of is .

Step 3: Detailed Explanation:
1. Find :
Given in .
2. Find coefficient in :
Since , the expression is .
The general term is .
For , :


Step 4: Final Answer
The coefficient of is .

Step 1: Understanding the Concept
A term independent of is a term where the total power of is zero. We first simplify the given expression into a single binomial form.

Step 2: Key Formula or Approach
1. Simplify . 2. Use the general term formula for , which is .

Step 3: Detailed Explanation
1. Rewrite the second bracket: 2. Combine the terms: 3. To find the term independent of , we need the coefficient of in the numerator , because . 4. The coefficient of in is . 5. Here, and . Thus, the coefficient is .

Step 4: Final Answer
The term independent of is .