Evaluate : `""^10C_1 + ""^10C_2 + ""^10C_3 + …. + ""^10C_10 `

Answer

To evaluate the sum \( \binom{10}{1} + \binom{10}{2} + \binom{10}{3} + \ldots + \binom{10}{10} \), we can use the properties of binomial coefficients.

### Step-by-Step Solution:

1. **Understand the Binomial Theorem**: The Binomial Theorem states that:
\[ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \]
For our case, we can set \( n = 10 \), \( x = 1 \), and \( y = 1 \).

2. **Apply the Binomial Theorem**: By substituting \( x = 1 \) and \( y = 1 \) into the theorem, we get:
\[ (1 + 1)^{10} = \sum_{k=0}^{10} \binom{10}{k} 1^{10-k} 1^k = \sum_{k=0}^{10} \binom{10}{k} \]
This simplifies to:
\[ 2^{10} = \sum_{k=0}^{10} \binom{10}{k} \]

3. **Calculate \( 2^{10} \)**: We know that:
\[ 2^{10} = 1024 \]

4. **Separate the Terms**: The sum \( \sum_{k=0}^{10} \binom{10}{k} \) includes all coefficients from \( \binom{10}{0} \) to \( \binom{10}{10} \). To find the sum from \( k=1 \) to \( k=10 \), we can subtract \( \binom{10}{0} \):
\[ \sum_{k=1}^{10} \binom{10}{k} = \sum_{k=0}^{10} \binom{10}{k} - \binom{10}{0} = 1024 - 1 \]

5. **Final Calculation**: Thus, we have:
\[ \sum_{k=1}^{10} \binom{10}{k} = 1024 - 1 = 1023 \]

### Conclusion:
The value of \( \binom{10}{1} + \binom{10}{2} + \binom{10}{3} + \ldots + \binom{10}{10} \) is \( 1023 \).