If `c_r=nc_r` then `(C_1)/2-(C_2)/3+(C_3)/4-...........-(C_100)/101` is equal to

Options

A. `C_(1)`
B. `C_(2)`
C. `C_(3)`
D. `C_(4)`

Answer

To solve the problem, we need to evaluate the expression:

\[ \frac{C_1}{2} - \frac{C_2}{3} + \frac{C_3}{4} - \ldots - \frac{C_{100}}{101} \]

where \(C_r\) represents the binomial coefficient \(\binom{100}{r}\).

### Step-by-Step Solution:

1. **Understanding the Binomial Coefficient**:
We know that \(C_r = \binom{100}{r}\), which represents the coefficients in the expansion of \((1 + x)^{100}\).

2. **Consider the Binomial Expansion**:
The binomial expansion of \((1 - x)^{100}\) is given by:
\[ (1 - x)^{100} = \sum_{r=0}^{100} \binom{100}{r} (-x)^r \]
This expands to:
\[ 1 - \binom{100}{1} x + \binom{100}{2} x^2 - \binom{100}{3} x^3 + \ldots + (-1)^{100} x^{100} \]

3. **Integrate the Expansion**:
We will integrate the expansion from \(0\) to \(1\):
\[ \int_0^1 (1 - x)^{100} \, dx \]

4. **Calculating the Integral**:
The integral can be calculated as follows:
\[ \int (1 - x)^{100} \, dx = -\frac{(1 - x)^{101}}{101} + C \]
Evaluating from \(0\) to \(1\):
\[ \left[-\frac{(1 - x)^{101}}{101}\right]_0^1 = \left[-\frac{0^{101}}{101}\right] - \left[-\frac{1^{101}}{101}\right] = 0 + \frac{1}{101} = \frac{1}{101} \]

5. **Setting Up the Result**:
The integral also corresponds to the sum of the series:
\[ \int_0^1 (1 - x)^{100} \, dx = \sum_{r=0}^{100} \binom{100}{r} \frac{(-1)^r}{r+1} \]
This means:
\[ \frac{C_0}{1} - \frac{C_1}{2} + \frac{C_2}{3} - \frac{C_3}{4} + \ldots + (-1)^{100} \frac{C_{100}}{101} = \frac{1}{101} \]

6. **Extracting the Required Expression**:
We need to isolate the expression:
\[ -\left(\frac{C_1}{2} - \frac{C_2}{3} + \frac{C_3}{4} - \ldots - \frac{C_{100}}{101}\right) \]
Rearranging gives:
\[ C_0 - \left(\frac{C_1}{2} - \frac{C_2}{3} + \frac{C_3}{4} - \ldots - \frac{C_{100}}{101}\right) = \frac{1}{101} \]

7. **Finding \(C_0\)**:
Here, \(C_0 = \binom{100}{0} = 1\).

8. **Final Calculation**:
Thus, we have:
\[ 1 - \left(\frac{C_1}{2} - \frac{C_2}{3} + \frac{C_3}{4} - \ldots - \frac{C_{100}}{101}\right) = \frac{1}{101} \]
Rearranging gives:
\[ \frac{C_1}{2} - \frac{C_2}{3} + \frac{C_3}{4} - \ldots - \frac{C_{100}}{101} = 1 - \frac{1}{101} = \frac{100}{101} \]

### Conclusion:
The value of the expression is:
\[ \frac{100}{101} \]