What is `C_0 + C_1 +…………..+C_n ` in `(1+x)^n`

Answer

To find the value of \( C_0 + C_1 + \ldots + C_n \) in the expansion of \( (1+x)^n \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Binomial Expansion**:
The binomial theorem states that:
\[ (1+x)^n = \sum_{r=0}^{n} C(n, r) x^r \]
where \( C(n, r) \) is the binomial coefficient, also denoted as \( nCr \) or \( \binom{n}{r} \).

2. **Identifying the Coefficients**:
The coefficients \( C(n, r) \) in the expansion correspond to \( C_0, C_1, \ldots, C_n \). Thus, we can express:
\[ C_0 + C_1 + \ldots + C_n = \sum_{r=0}^{n} C(n, r) \]

3. **Substituting \( x = 1 \)**:
To find the sum of the coefficients, we can substitute \( x = 1 \) in the binomial expansion:
\[ (1+1)^n = \sum_{r=0}^{n} C(n, r) \cdot 1^r \]
This simplifies to:
\[ 2^n = C_0 + C_1 + \ldots + C_n \]

4. **Conclusion**:
Therefore, we conclude that:
\[ C_0 + C_1 + \ldots + C_n = 2^n \]

### Final Answer:
\[ C_0 + C_1 + \ldots + C_n = 2^n \]