Find the value of `^10C_1 + ^10C_2 + ^10C_3 + ... + ^10C_9`

Options

A. 100
B. `10^2 - 10`
C. `10^2 - 11`
D. `2(2^9 -1)`

Answer

To find the value of \( ^{10}C_1 + ^{10}C_2 + ^{10}C_3 + ... + ^{10}C_9 \), we can use the properties of combinations and the binomial theorem. Here’s a step-by-step solution:

### Step 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.

### Step 2: Apply the Binomial Theorem for \( n = 10 \)
For our case, we set \( n = 10 \):
\[ (1 + x)^{10} = \sum_{k=0}^{10} {10 \choose k} x^k \]

### Step 3: Substitute \( x = 1 \)
Now, substituting \( x = 1 \) into the equation gives:
\[ (1 + 1)^{10} = \sum_{k=0}^{10} {10 \choose k} (1)^k \]
This simplifies to:
\[ 2^{10} = {10 \choose 0} + {10 \choose 1} + {10 \choose 2} + \ldots + {10 \choose 10} \]

### Step 4: Calculate \( 2^{10} \)
Calculating \( 2^{10} \):
\[ 2^{10} = 1024 \]

### Step 5: Separate the Terms
We now know:
\[ 1024 = {10 \choose 0} + {10 \choose 1} + {10 \choose 2} + \ldots + {10 \choose 10} \]
We can separate the terms:
\[ 1024 = 1 + {10 \choose 1} + {10 \choose 2} + \ldots + {10 \choose 9} + {10 \choose 10} \]

### Step 6: Recognize the Symmetry in Combinations
Notice that \( {10 \choose 10} = 1 \) and \( {10 \choose 0} = 1 \). Thus, we can rewrite:
\[ {10 \choose 1} + {10 \choose 2} + \ldots + {10 \choose 9} = 1024 - 1 - 1 = 1022 \]

### Step 7: Final Calculation
Thus, the value of \( ^{10}C_1 + ^{10}C_2 + ^{10}C_3 + ... + ^{10}C_9 \) is:
\[ ^{10}C_1 + ^{10}C_2 + \ldots + ^{10}C_9 = 1022 \]

### Final Answer
The final answer is:
\[ 1022 \]