Evaluate `""^(10)C_1 + ""^(10)C_2 + ""^(10)C_3 + ………+""^10C_10`

Answer

To evaluate the expression \( \binom{10}{1} + \binom{10}{2} + \binom{10}{3} + \ldots + \binom{10}{10} \), we can use the Binomial Theorem.

### Step-by-Step Solution:

1. **Understanding the Binomial Theorem**:
The Binomial Theorem states that:
\[ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \]
For \( n = 10 \), this becomes:
\[ (x + y)^{10} = \sum_{k=0}^{10} \binom{10}{k} x^{10-k} y^k \]

2. **Setting Values for x and y**:
To find the sum \( \binom{10}{1} + \binom{10}{2} + \binom{10}{3} + \ldots + \binom{10}{10} \), we can set \( x = 1 \) and \( y = 1 \):
\[ (1 + 1)^{10} = \sum_{k=0}^{10} \binom{10}{k} 1^{10-k} 1^k = \sum_{k=0}^{10} \binom{10}{k} \]

3. **Calculating the Left Side**:
The left side simplifies to:
\[ 2^{10} = 1024 \]

4. **Identifying the Terms**:
The sum \( \sum_{k=0}^{10} \binom{10}{k} \) includes all the binomial coefficients from \( k = 0 \) to \( k = 10 \). We want to find:
\[ \binom{10}{1} + \binom{10}{2} + \binom{10}{3} + \ldots + \binom{10}{10} \]
This can be expressed as:
\[ \sum_{k=1}^{10} \binom{10}{k} = \sum_{k=0}^{10} \binom{10}{k} - \binom{10}{0} \]

5. **Subtracting the Zero Term**:
Since \( \binom{10}{0} = 1 \):
\[ \sum_{k=1}^{10} \binom{10}{k} = 1024 - 1 = 1023 \]

### Final Answer:
Thus, the value of \( \binom{10}{1} + \binom{10}{2} + \binom{10}{3} + \ldots + \binom{10}{10} \) is \( \boxed{1023} \).