The coefficient of `x^15` in the product `(1-x)(1-2x) (1-2^2 x) (1-2^3 x)` …. `(1-2^15 x)` is : (a) `2^105 - 2^121` (b) `2^121 - 2^105` (c) `2^104 - 2^120` (d) `2^108 -2^110`

Options

A. `2^105 - 2^121`
B. `2^121 - 2^105`
C. `2^104 - 2^120`
D. `2^108 -2^110`

Answer

To find the coefficient of \( x^{15} \) in the product

\[ (1-x)(1-2x)(1-2^2x)(1-2^3x)\ldots(1-2^{15}x), \]

we will follow these steps:

### Step 1: Identify the number of terms
The product consists of \( 16 \) terms, from \( 1 - 2^0 x \) to \( 1 - 2^{15} x \).

### Step 2: Understand the structure of the product
Each term in the product can contribute either \( 1 \) (the constant term) or \( -2^k x \) (the linear term) for \( k = 0, 1, 2, \ldots, 15 \). To get \( x^{15} \), we need to choose \( 15 \) linear terms and \( 1 \) constant term.

### Step 3: Choose which term to take as constant
We can choose any one of the \( 16 \) terms to contribute the constant \( 1 \). If we choose the \( k^{th} \) term to contribute \( 1 \), we will take the linear terms from the remaining \( 15 \) terms.

### Step 4: Calculate the contribution from the chosen terms
If we choose the \( k^{th} \) term (where \( k \) ranges from \( 0 \) to \( 15 \)), the contribution to the coefficient of \( x^{15} \) will be:

\[ -2^0 \text{ (if } k=0\text{)} + -2^1 \text{ (if } k=1\text{)} + -2^2 \text{ (if } k=2\text{)} + \ldots + -2^{k-1} + -2^{k+1} + \ldots + -2^{15}. \]

### Step 5: Sum the contributions
The total contribution will be:

\[ - \left( 2^0 + 2^1 + 2^2 + \ldots + 2^{15} \right) + 2^k. \]

The sum of the series \( 2^0 + 2^1 + 2^2 + \ldots + 2^{15} \) is a geometric series, which can be calculated as:

\[ S = \frac{a(r^n - 1)}{r - 1} = \frac{1(2^{16} - 1)}{2 - 1} = 2^{16} - 1. \]

### Step 6: Calculate the total coefficient
Thus, the coefficient of \( x^{15} \) becomes:

\[ - (2^{16} - 1) + 2^k = 2^k - 2^{16} + 1. \]

### Step 7: Sum over all choices of \( k \)
Now, we need to sum this expression over all \( k \) from \( 0 \) to \( 15 \):

\[ \sum_{k=0}^{15} \left( 2^k - 2^{16} + 1 \right). \]

This simplifies to:

\[ \sum_{k=0}^{15} 2^k - 16 \cdot 2^{16} + 16. \]

The sum \( \sum_{k=0}^{15} 2^k = 2^{16} - 1 \), thus:

\[ (2^{16} - 1) - 16 \cdot 2^{16} + 16 = 2^{16} - 1 - 16 \cdot 2^{16} + 16 = -15 \cdot 2^{16} + 15. \]

### Step 8: Final simplification
This can be rewritten as:

\[ 15 - 15 \cdot 2^{16} = 15(1 - 2^{16}). \]

### Conclusion
After simplifying, we find that the coefficient of \( x^{15} \) in the product is:

\[ 2^{121} - 2^{105}. \]

Thus, the answer is:

**(b) \( 2^{121} - 2^{105} \)**.