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)-.^(16)C_(5)`
C. `.^(16)C_(6)-1`
D. `.^(16)C_(6)`

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 individual expansions
The expression consists of the sum of several binomial expansions:
\[ (1+x)^6, (1+x)^7, \ldots, (1+x)^{15} \]
We need to find the coefficient of \( x^6 \) in each of these expansions.

### Step 2: Find the coefficient of \( x^6 \) in each term
The coefficient of \( x^6 \) in the expansion of \( (1+x)^n \) is given by \( \binom{n}{6} \). Therefore, we need to calculate:
\[ \binom{6}{6}, \binom{7}{6}, \binom{8}{6}, \ldots, \binom{15}{6} \]

### Step 3: Write out the coefficients
Calculating these coefficients, we have:
- \( \binom{6}{6} = 1 \)
- \( \binom{7}{6} = 7 \)
- \( \binom{8}{6} = 28 \)
- \( \binom{9}{6} = 84 \)
- \( \binom{10}{6} = 210 \)
- \( \binom{11}{6} = 462 \)
- \( \binom{12}{6} = 924 \)
- \( \binom{13}{6} = 1716 \)
- \( \binom{14}{6} = 3003 \)
- \( \binom{15}{6} = 5005 \)

### Step 4: Sum the coefficients
Now, we sum all these coefficients:
\[ 1 + 7 + 28 + 84 + 210 + 462 + 924 + 1716 + 3003 + 5005 \]

### Step 5: Calculate the total
Calculating the total:
\[ 1 + 7 = 8 \\ 8 + 28 = 36 \\ 36 + 84 = 120 \\ 120 + 210 = 330 \\ 330 + 462 = 792 \\ 792 + 924 = 1716 \\ 1716 + 1716 = 3432 \\ 3432 + 3003 = 6435 \\ 6435 + 5005 = 11440 \]

### Final Result
The coefficient of \( x^6 \) in the expression \( (1+x)^6 + (1+x)^7 + \ldots + (1+x)^{15} \) is:
\[ \boxed{11440} \]