Coefficient of `x^(17)` in the polynomial `P(x)=prod_(r=0)^(17)(x+""^(35)C_(r))` is

Options

A. `2^(34)`
B. `""^(36)C_(17)`
C. `2^(35)-""^(36)C_(17)`
D. 0

Answer

To find the coefficient of \( x^{17} \) in the polynomial

\[ P(x) = \prod_{r=0}^{17} \left( x + \binom{35}{r} \right), \]

we can follow these steps:

### Step 1: Expand the Product
The polynomial \( P(x) \) can be expanded as follows:

\[ P(x) = (x + \binom{35}{0})(x + \binom{35}{1})(x + \binom{35}{2}) \cdots (x + \binom{35}{17}). \]

### Step 2: Identify the Terms Contributing to \( x^{17} \)
To find the coefficient of \( x^{17} \), we need to consider the terms that will yield \( x^{17} \) when multiplied together. This occurs when we choose \( x \) from 17 of the factors and the constant term (the binomial coefficient) from one of the factors.

### Step 3: Choose One Constant Term
We can choose one constant term from one of the factors and \( x \) from the remaining 17 factors. The constant term we can choose from the \( r \)-th factor is \( \binom{35}{r} \).

### Step 4: Calculate the Coefficient
The coefficient of \( x^{17} \) in \( P(x) \) is the sum of the binomial coefficients from \( r = 0 \) to \( r = 17 \):

\[ \text{Coefficient of } x^{17} = \sum_{r=0}^{17} \binom{35}{r}. \]

### Step 5: Use the Binomial Theorem
According to the Binomial Theorem, we know that:

\[ \sum_{r=0}^{n} \binom{n}{r} = 2^n. \]

For \( n = 35 \):

\[ \sum_{r=0}^{35} \binom{35}{r} = 2^{35}. \]

### Step 6: Relate to the Required Sum
Since we need the sum from \( r = 0 \) to \( r = 17 \), we can use the symmetry property of binomial coefficients:

\[ \binom{35}{r} = \binom{35}{35-r}. \]

Thus, the sum of the coefficients from \( r = 0 \) to \( r = 17 \) is equal to the sum from \( r = 18 \) to \( r = 35 \). Therefore:

\[ \sum_{r=0}^{17} \binom{35}{r} = \sum_{r=18}^{35} \binom{35}{r} = \frac{1}{2} \sum_{r=0}^{35} \binom{35}{r} = \frac{1}{2} \cdot 2^{35} = 2^{34}. \]

### Final Result
Thus, the coefficient of \( x^{17} \) in the polynomial \( P(x) \) is:

\[ \boxed{2^{34}}. \]