Let `n` be a positive even integer. The ratio of the largest coefficient and the `2^(nd)` largest coefficient in the expansion of `(1+x)^(n)` is `11:10.` Then the number of terms in the expansion of `(1+x)^(n)` is

Options

A. 20
B. 21
C. 10
D. 11

Answer

To solve the problem step by step, we will analyze the given information and apply the concepts of the Binomial Theorem.

### Step 1: Understanding the Problem
We are given that \( n \) is a positive even integer. The ratio of the largest coefficient to the second largest coefficient in the expansion of \( (1+x)^n \) is \( \frac{11}{10} \).

### Step 2: Identify the Largest Coefficient
For the expansion of \( (1+x)^n \), the coefficients are given by \( \binom{n}{k} \) for \( k = 0, 1, 2, \ldots, n \). The largest coefficient occurs at \( k = \frac{n}{2} \) when \( n \) is even. Therefore, the largest coefficient is:
\[ \text{Largest Coefficient} = \binom{n}{\frac{n}{2}} = \binom{n}{m} \quad \text{(where \( n = 2m \))} \]

### Step 3: Identify the Second Largest Coefficient
The second largest coefficient occurs at \( k = \frac{n}{2} - 1 \):
\[ \text{Second Largest Coefficient} = \binom{n}{\frac{n}{2} - 1} = \binom{n}{m-1} \]

### Step 4: Set Up the Ratio
We know from the problem that:
\[ \frac{\text{Largest Coefficient}}{\text{Second Largest Coefficient}} = \frac{11}{10} \]
Substituting the coefficients we found:
\[ \frac{\binom{n}{m}}{\binom{n}{m-1}} = \frac{11}{10} \]

### Step 5: Simplify the Ratio of Binomial Coefficients
Using the property of binomial coefficients:
\[ \frac{\binom{n}{m}}{\binom{n}{m-1}} = \frac{m}{n - m + 1} \]
Substituting \( n = 2m \):
\[ \frac{m}{2m - m + 1} = \frac{m}{m + 1} \]
Setting this equal to \( \frac{11}{10} \):
\[ \frac{m}{m + 1} = \frac{11}{10} \]

### Step 6: Cross Multiply and Solve for \( m \)
Cross multiplying gives:
\[ 10m = 11(m + 1) \]
Expanding and rearranging:
\[ 10m = 11m + 11 \implies 11 = 11m - 10m \implies m = 11 \]

### Step 7: Find \( n \)
Since \( n = 2m \):
\[ n = 2 \times 11 = 22 \]

### Step 8: Determine the Number of Terms
The number of terms in the expansion of \( (1+x)^n \) is given by \( n + 1 \):
\[ \text{Number of Terms} = n + 1 = 22 + 1 = 23 \]

### Final Answer
Thus, the number of terms in the expansion of \( (1+x)^n \) is \( \boxed{23} \).