Find the coefficient of `x^49` in the polynomial `(x-C_1/C_2)(x-2^2*C-2/C_1)(x-3^2*C_3/c_2)............(x-50^2*c_50/C-49)`, where `C_r=50C_r`.

Answer

To find the coefficient of \( x^{49} \) in the polynomial

\[ P(x) = (x - \frac{C_1}{C_2})(x - \frac{2^2 C_2}{C_1})(x - \frac{3^2 C_3}{C_2}) \ldots (x - \frac{50^2 C_{50}}{C_{49}}) \]

where \( C_r = \binom{50}{r} \), we will follow these steps:

### Step 1: Identify the structure of the polynomial
The polynomial \( P(x) \) consists of 50 linear factors. Each factor is of the form \( (x - a_i) \) where \( a_i = \frac{i^2 C_i}{C_{i-1}} \) for \( i = 1, 2, \ldots, 50 \).

### Step 2: Find the values of \( a_i \)
We need to calculate \( a_i \) for \( i = 1, 2, \ldots, 50 \):
- \( C_1 = \binom{50}{1} = 50 \)
- \( C_2 = \binom{50}{2} = \frac{50 \times 49}{2} = 1225 \)
- \( C_3 = \binom{50}{3} = \frac{50 \times 49 \times 48}{6} = 19600 \)
- Continuing this way, we can find \( C_i \) for \( i = 1 \) to \( 50 \).

Now, we can compute \( a_i \):
- \( a_1 = \frac{1^2 \cdot 50}{1} = 50 \)
- \( a_2 = \frac{2^2 \cdot 1225}{50} = 98 \)
- \( a_3 = \frac{3^2 \cdot 19600}{1225} = 144 \)
- Continuing this way, we can find \( a_i \) for \( i = 1 \) to \( 50 \).

### Step 3: Coefficient of \( x^{49} \)
The coefficient of \( x^{49} \) in \( P(x) \) is the sum of the products of the roots taken one at a time (with a negative sign). This is given by:

\[ - \sum_{i=1}^{50} a_i \]

### Step 4: Calculate the sum of \( a_i \)
We need to compute the sum \( \sum_{i=1}^{50} a_i \):
- Using the values calculated in Step 2, we can sum them up.

### Step 5: Final coefficient
The coefficient of \( x^{49} \) is then:

\[ - \sum_{i=1}^{50} a_i \]

### Conclusion
After calculating the sum of \( a_i \) values, we will arrive at the final coefficient of \( x^{49} \).