If the coefficient of `r^(th)` and `(r+4)^(th)` terms are equal in the expansion of `(1+x)^(20)`, then the value of `r` will be

Options

A. (a) 7
B. (b) 8
C. (c) 9
D. (d) 10

Answer

To solve the problem, we need to find the value of \( r \) such that the coefficients of the \( r^{th} \) and \( (r+4)^{th} \) terms in the expansion of \( (1+x)^{20} \) are equal.

### Step-by-Step Solution:

1. **Identify the General Term:**
The general term \( T_{r+1} \) in the expansion of \( (1+x)^{20} \) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
Here, \( n = 20 \), \( a = 1 \), and \( b = x \). Thus, the general term simplifies to:
\[ T_{r+1} = \binom{20}{r} x^r \]

2. **Find the Coefficient of the \( r^{th} \) Term:**
The coefficient of the \( r^{th} \) term (which corresponds to \( T_{r} \)) is:
\[ \text{Coefficient of } T_r = \binom{20}{r} \]

3. **Find the Coefficient of the \( (r+4)^{th} \) Term:**
The coefficient of the \( (r+4)^{th} \) term (which corresponds to \( T_{r+4} \)) is:
\[ \text{Coefficient of } T_{r+4} = \binom{20}{r+4} \]

4. **Set the Coefficients Equal:**
According to the problem, these coefficients are equal:
\[ \binom{20}{r} = \binom{20}{r+4} \]

5. **Use the Property of Binomial Coefficients:**
The property of binomial coefficients states that:
\[ \binom{n}{k} = \binom{n}{n-k} \]
Therefore, we can write:
\[ \binom{20}{r} = \binom{20}{20 - (r + 4)} = \binom{20}{16 - r} \]

6. **Set the Indices Equal:**
Since the coefficients are equal, we can set the indices equal to each other:
\[ r = 16 - r \]

7. **Solve for \( r \):**
Rearranging the equation gives:
\[ r + r = 16 \]
\[ 2r = 16 \]
\[ r = 8 \]

### Conclusion:
The value of \( r \) is \( 8 \).