Find the value of `^10C_1 + ^10C_2 + ^10C_3 + ... + ^10C_10`

Options

A. 100
B. `10^2`
C. `2^10`
D. `2^10 - 1`

Answer

To find the value of \( ^{10}C_1 + ^{10}C_2 + ^{10}C_3 + ... + ^{10}C_{10} \), we can use the properties of binomial coefficients.

### Step-by-Step Solution:

1. **Understand the Binomial Theorem**: The Binomial Theorem states that:
\[ (1 + x)^n = \sum_{k=0}^{n} {n \choose k} x^k \]
where \( {n \choose k} \) is the binomial coefficient.

2. **Set \( n = 10 \)**: In our case, we want to find the sum of the binomial coefficients from \( k = 1 \) to \( k = 10 \) for \( n = 10 \). Thus, we can express it as:
\[ (1 + x)^{10} = {10 \choose 0} + {10 \choose 1} x + {10 \choose 2} x^2 + \ldots + {10 \choose 10} x^{10} \]

3. **Substitute \( x = 1 \)**: By substituting \( x = 1 \) into the equation, we get:
\[ (1 + 1)^{10} = 2^{10} = {10 \choose 0} + {10 \choose 1} + {10 \choose 2} + \ldots + {10 \choose 10} \]
This simplifies to:
\[ 2^{10} = {10 \choose 0} + \sum_{k=1}^{10} {10 \choose k} \]

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

5. **Isolate the Sum**: Since \( {10 \choose 0} = 1 \), we can rewrite the equation:
\[ 1024 = 1 + \sum_{k=1}^{10} {10 \choose k} \]
Therefore,
\[ \sum_{k=1}^{10} {10 \choose k} = 1024 - 1 = 1023 \]

6. **Final Result**: Thus, the value of \( ^{10}C_1 + ^{10}C_2 + ^{10}C_3 + ... + ^{10}C_{10} \) is:
\[ \boxed{1023} \]