In the binomial expansion of `(1 + x)^(10)`, the coefficeents of `(2m + 1)^(th) and (4m + 5)^(th)` terms are equal. Value of m is

Options

A. -1
B. 2
C. 3
D. None of these

Answer

To solve the problem, we need to find the value of \( m \) such that the coefficients of the \( (2m + 1)^{th} \) term and the \( (4m + 5)^{th} \) term in the binomial expansion of \( (1 + x)^{10} \) are equal.

### Step-by-Step Solution:

1. **Identify the General Term**:
The \( r^{th} \) term in the binomial expansion of \( (1 + x)^n \) is given by:
\[ T_r = \binom{n}{r-1} x^{r-1} \]
For our case, \( n = 10 \), so:
\[ T_r = \binom{10}{r-1} x^{r-1} \]

2. **Coefficients of the Terms**:
- The coefficient of the \( (2m + 1)^{th} \) term is:
\[ \text{Coefficient of } T_{2m + 1} = \binom{10}{(2m + 1) - 1} = \binom{10}{2m} \]
- The coefficient of the \( (4m + 5)^{th} \) term is:
\[ \text{Coefficient of } T_{4m + 5} = \binom{10}{(4m + 5) - 1} = \binom{10}{4m + 4} \]

3. **Setting the Coefficients Equal**:
We need to set the coefficients equal to each other:
\[ \binom{10}{2m} = \binom{10}{4m + 4} \]

4. **Using the Property of Binomial Coefficients**:
The property of binomial coefficients states that:
\[ \binom{n}{r} = \binom{n}{n - r} \]
Therefore, we can write:
\[ \binom{10}{4m + 4} = \binom{10}{10 - (4m + 4)} = \binom{10}{6 - 4m} \]

5. **Setting the Indices Equal**:
This gives us two cases to consider:
- Case 1:
\[ 2m = 4m + 4 \]
- Case 2:
\[ 2m = 6 - 4m \]

6. **Solving Case 1**:
From \( 2m = 4m + 4 \):
\[ 2m - 4m = 4 \implies -2m = 4 \implies m = -2 \]
(This value is not valid as \( m \) cannot be negative.)

7. **Solving Case 2**:
From \( 2m = 6 - 4m \):
\[ 2m + 4m = 6 \implies 6m = 6 \implies m = 1 \]

8. **Conclusion**:
The only valid solution is:
\[ m = 1 \]

### Final Answer:
The value of \( m \) is \( 1 \).