The coefficient of `x^6` in `{(1+x)^6+(1+x)^7+........+(1+x)^(15)}` is

Options

A. `""^(16)C_(9)`
B. `""^(16)C_(5)-""^(6)C_(5)`
C. `""^(16)C_(6) -1`
D. none of these

Answer

To find the coefficient of \( x^6 \) in the expression \( (1+x)^6 + (1+x)^7 + \ldots + (1+x)^{15} \), we can follow these steps:

### Step 1: Identify the Series
The expression can be seen as a sum of powers of \( (1+x) \) from 6 to 15. This can be rewritten as:
\[ S = (1+x)^6 + (1+x)^7 + (1+x)^8 + \ldots + (1+x)^{15} \]

### Step 2: Recognize the Geometric Progression
This series is a geometric progression (GP) where:
- The first term \( a = (1+x)^6 \)
- The common ratio \( r = (1+x) \)
- The number of terms \( n = 15 - 6 + 1 = 10 \)

### Step 3: Use the Formula for the Sum of a GP
The sum of a GP can be calculated using the formula:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
Substituting the values we have:
\[ S = (1+x)^6 \frac{(1+x)^{10} - 1}{(1+x) - 1} \]
This simplifies to:
\[ S = (1+x)^6 \frac{(1+x)^{10} - 1}{x} \]

### Step 4: Expand the Expression
Now we can expand the expression:
\[ S = \frac{(1+x)^{16} - (1+x)^6}{x} \]

### Step 5: Find the Coefficient of \( x^6 \)
To find the coefficient of \( x^6 \) in \( S \), we need to consider two parts:
1. The term \( \frac{(1+x)^{16}}{x} \) contributes to \( x^6 \) when we find the coefficient of \( x^7 \) in \( (1+x)^{16} \).
2. The term \( -\frac{(1+x)^6}{x} \) does not contribute to \( x^6 \) since it will give terms of lower power.

Thus, we need to find the coefficient of \( x^7 \) in \( (1+x)^{16} \).

### Step 6: Calculate the Coefficient
The coefficient of \( x^7 \) in \( (1+x)^{16} \) is given by:
\[ \binom{16}{7} \]

### Step 7: Final Answer
Thus, the coefficient of \( x^6 \) in the original expression is:
\[ \binom{16}{7} = \binom{16}{9} \]
So, the answer is \( \binom{16}{9} \).