Given the integers `r gt 1, n gt 2` and coefficients of `(3r)` th and `(r+2)` th terms in the expansion of `(1+x)^(2n)` are equal, then

Options

A. `n=2r`
B. `n=3r`
C. `n=2r+1`
D. `2r-1`

Answer

To solve the problem, we need to find the relationship between the integers \( n \) and \( r \) based on the given condition that the coefficients of the \( (3r) \)-th term and the \( (r+2) \)-th term in the expansion of \( (1+x)^{2n} \) are equal.

### Step-by-Step Solution:

1. **Identify the Coefficient Formula**:
The coefficient of the \( k \)-th term in the expansion of \( (1+x)^{2n} \) is given by \( \binom{2n}{k} \). Therefore, the coefficients of the \( (3r) \)-th term and the \( (r+2) \)-th term are:
- Coefficient of \( (3r) \)-th term: \( \binom{2n}{3r} \)
- Coefficient of \( (r+2) \)-th term: \( \binom{2n}{r+2} \)

2. **Set the Coefficients Equal**:
According to the problem, these coefficients are equal:
\[ \binom{2n}{3r} = \binom{2n}{r+2} \]

3. **Use the Property of Binomial Coefficients**:
The property of binomial coefficients states that if \( \binom{n}{k} = \binom{n}{m} \), then \( k + m = n \). Here, we can apply this property:
\[ 3r + (r + 2) = 2n \]

4. **Simplify the Equation**:
Combine the terms on the left side:
\[ 3r + r + 2 = 2n \implies 4r + 2 = 2n \]

5. **Divide by 2**:
To isolate \( n \), divide the entire equation by 2:
\[ 2r + 1 = n \]

6. **Final Result**:
Thus, we have established the relationship between \( n \) and \( r \):
\[ n = 2r + 1 \]