The coefficient of `x^50` in the polynomial `(x + ^50C_0)(x +3.^5C_1) (x +5.^5C_2).....(x + (2n + 1) ^5C_50)`, is

Options

A. `50.2^(50)`
B. `50.2^(51)`
C. `51.2^(50)`
D. `50 .(2^(50) +1)`

Explanation

To find the coefficient of \( x^{50} \) in the polynomial

\[ (x + \binom{50}{0})(x + 3 \cdot \binom{50}{1})(x + 5 \cdot \binom{50}{2}) \ldots (x + (2n + 1) \cdot \binom{50}{50}), \]

we will follow these steps:

### Step 1: Identify the structure of the polynomial
The polynomial consists of terms of the form \( (x + (2r + 1) \cdot \binom{50}{r}) \) for \( r = 0 \) to \( 50 \). The maximum power of \( x \) in this polynomial is \( 51 \) (when we take \( x \) from each term).

### Step 2: Coefficient of \( x^{50} \)
The coefficient of \( x^{50} \) can be found by considering the contributions from the terms where we take \( x \) from 50 of the factors and the constant term from one factor.

### Step 3: Write the expression for the coefficient
The coefficient of \( x^{50} \) is given by:

\[ - \left( \binom{50}{0} + 3 \cdot \binom{50}{1} + 5 \cdot \binom{50}{2} + \ldots + (2 \cdot 50 + 1) \cdot \binom{50}{50} \right). \]

### Step 4: General term representation
The general term can be expressed as:

\[ \sum_{r=0}^{50} (2r + 1) \cdot \binom{50}{r}. \]

### Step 5: Split the summation
We can split the summation into two parts:

\[ \sum_{r=0}^{50} (2r + 1) \cdot \binom{50}{r} = \sum_{r=0}^{50} 2r \cdot \binom{50}{r} + \sum_{r=0}^{50} \binom{50}{r}. \]

### Step 6: Evaluate the second summation
The second summation is simply \( 2^{50} \) because it represents the sum of the binomial coefficients.

### Step 7: Evaluate the first summation
For the first summation, we can use the identity \( r \cdot \binom{n}{r} = n \cdot \binom{n-1}{r-1} \):

\[ \sum_{r=0}^{50} 2r \cdot \binom{50}{r} = 2 \cdot 50 \cdot \sum_{r=1}^{50} \binom{49}{r-1} = 100 \cdot 2^{49}. \]

### Step 8: Combine the results
Now we combine the results:

\[ \sum_{r=0}^{50} (2r + 1) \cdot \binom{50}{r} = 100 \cdot 2^{49} + 2^{50} = 100 \cdot 2^{49} + 2 \cdot 2^{49} = 102 \cdot 2^{49}. \]

### Step 9: Find the coefficient of \( x^{50} \)
Thus, the coefficient of \( x^{50} \) is:

\[ - \left( 102 \cdot 2^{49} \right). \]

### Final Answer
The coefficient of \( x^{50} \) in the polynomial is:

\[ -102 \cdot 2^{49}. \]