Evaluate :
`1+^(15)C_(1)+^(15)C_(2)+^(15)C_(3)+......+^(15)C_(15)`

Options

A. `2^14`
B. `2^15`
C. `2^16`
D. `2^13`

Explanation

To evaluate the expression \(1 + \binom{15}{1} + \binom{15}{2} + \binom{15}{3} + \ldots + \binom{15}{15}\), we can use the Binomial Theorem.

### Step-by-step Solution:

1. **Understanding the Binomial Theorem**: The Binomial Theorem states that:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
For our case, we can set \(a = 1\) and \(b = 1\), and \(n = 15\).

2. **Applying the Binomial Theorem**: Substitute \(a\) and \(b\) into the theorem:
\[ (1 + 1)^{15} = \sum_{k=0}^{15} \binom{15}{k} 1^{15-k} 1^k \]
This simplifies to:
\[ 2^{15} = \sum_{k=0}^{15} \binom{15}{k} \]

3. **Identifying the Series**: The series we want to evaluate is:
\[ 1 + \binom{15}{1} + \binom{15}{2} + \binom{15}{3} + \ldots + \binom{15}{15} \]
Notice that this series is equivalent to:
\[ \sum_{k=0}^{15} \binom{15}{k} \]
which is the same as \(2^{15}\).

4. **Calculating \(2^{15}\)**: Now we calculate \(2^{15}\):
\[ 2^{15} = 32768 \]

5. **Final Result**: Therefore, the value of the expression \(1 + \binom{15}{1} + \binom{15}{2} + \ldots + \binom{15}{15}\) is:
\[ \boxed{32768} \]