`C_0^2 -C_1^2 + C_2^2 - ………-C_15^2` =

Options

A. 1
B. 2
C. 3
D. 0

Answer

To solve the expression \( C_0^2 - C_1^2 + C_2^2 - C_3^2 + \ldots - C_{15}^2 \), we will use the properties of binomial coefficients.

### Step-by-step Solution:

1. **Understanding the Binomial Coefficients**:
The binomial coefficient \( C_n^k \) represents the number of ways to choose \( k \) elements from a set of \( n \) elements. The coefficients can be calculated using the formula:
\[ C_n^k = \frac{n!}{k!(n-k)!} \]

2. **Recognizing the Pattern**:
The expression alternates between positive and negative signs. We can rewrite the expression as:
\[ \sum_{k=0}^{15} (-1)^k C_{15}^k \]

3. **Using the Binomial Theorem**:
According to the binomial theorem, the sum of the binomial coefficients with alternating signs can be expressed as:
\[ (1 - 1)^{15} = 0 \]
This is because \( (1 - 1)^{n} = 0 \) for any positive integer \( n \).

4. **Conclusion**:
Since \( n = 15 \) is odd, the sum \( C_0^2 - C_1^2 + C_2^2 - C_3^2 + \ldots - C_{15}^2 \) evaluates to:
\[ 0 \]

### Final Answer:
\[ C_0^2 - C_1^2 + C_2^2 - C_3^2 + \ldots - C_{15}^2 = 0 \]