Find `1/2.""^10C_0 -""^10C_1 +2.""^10C_2 - 2^2.""^10C_3+…..+2^9. ""^10C_10 = ` ?

Options

A. `1//2`
B. `1//4`
C. `3//2`
D. `1//3`

Answer

To solve the expression \( S = \frac{1}{2} \binom{10}{0} - \binom{10}{1} + 2 \binom{10}{2} - 2^2 \binom{10}{3} + \ldots + 2^9 \binom{10}{10} \), we will follow these steps:

### Step 1: Define the sum
Let us denote the sum as \( S \):
\[ S = \frac{1}{2} \binom{10}{0} - \binom{10}{1} + 2 \binom{10}{2} - 2^2 \binom{10}{3} + \ldots + 2^9 \binom{10}{10} \]

### Step 2: Multiply the sum by 2
Now, multiply the entire sum \( S \) by 2:
\[ 2S = 2 \left( \frac{1}{2} \binom{10}{0} - \binom{10}{1} + 2 \binom{10}{2} - 2^2 \binom{10}{3} + \ldots + 2^9 \binom{10}{10} \right) \]
This simplifies to:
\[ 2S = \binom{10}{0} - 2 \binom{10}{1} + 2^2 \binom{10}{2} - 2^3 \binom{10}{3} + \ldots + 2^{10} \binom{10}{10} \]

### Step 3: Recognize the binomial expansion
The right-hand side can be recognized as the expansion of \( (1 - 2)^{10} \):
\[ 2S = (1 - 2)^{10} = (-1)^{10} = 1 \]

### Step 4: Solve for \( S \)
Now, we can solve for \( S \):
\[ 2S = 1 \implies S = \frac{1}{2} \]

### Final Answer
Thus, the value of the original sum is:
\[ \boxed{\frac{1}{2}} \]