The coefficient of `x^1007` in the expansion `(1+x)^(2006)+x(1+x)^(2005)+x^2(1+x)^(2004)x^3(1+x)^(2003)+..... +x^(2006)` is

Options

A. `.^(2006)C_(1007)`
B. `.^(2006)C_(1006)`
C. `.^(2007)C_(1006)`
D. `.^(2007)C_(1007)`

Answer

To find the coefficient of \( x^{1007} \) in the expansion

\[ (1+x)^{2006} + x(1+x)^{2005} + x^2(1+x)^{2004} + \ldots + x^{2006}, \]

we can break down the problem step by step.

### Step 1: Rewrite the Expression
The expression can be rewritten as:

\[ \sum_{k=0}^{2006} x^k (1+x)^{2006-k}. \]

This means we are summing terms where \( k \) varies from 0 to 2006.

### Step 2: Identify the Coefficient of \( x^{1007} \)
To find the coefficient of \( x^{1007} \), we need to consider the contributions from each term in the sum. For each \( k \), the term \( x^k (1+x)^{2006-k} \) contributes to \( x^{1007} \) if:

\[ k + j = 1007, \]

where \( j \) is the power of \( x \) in the expansion of \( (1+x)^{2006-k} \). Thus, we have:

\[ j = 1007 - k. \]

### Step 3: Find the Range for \( k \)
Since \( j \) must be a non-negative integer, we require:

\[ 1007 - k \geq 0 \implies k \leq 1007. \]

However, \( k \) can only go up to 2006, so the maximum value of \( k \) is 1007.

### Step 4: Coefficients from the Binomial Expansion
The coefficient of \( x^j \) in \( (1+x)^{2006-k} \) is given by \( \binom{2006-k}{j} \). Therefore, we need to sum the coefficients for all valid \( k \):

\[ \text{Coefficient of } x^{1007} = \sum_{k=0}^{1007} \binom{2006-k}{1007-k}. \]

### Step 5: Change of Variables
Let \( m = 1007 - k \). Then, as \( k \) goes from 0 to 1007, \( m \) goes from 1007 to 0. Thus, we can rewrite the sum:

\[ \sum_{m=0}^{1007} \binom{2006 - (1007 - m)}{m} = \sum_{m=0}^{1007} \binom{999 + m}{m}. \]

### Step 6: Use the Hockey Stick Identity
Using the hockey stick identity in combinatorics, we have:

\[ \sum_{i=r}^{n} \binom{i}{r} = \binom{n+1}{r+1}. \]

In our case, we apply it as follows:

\[ \sum_{m=0}^{1007} \binom{999+m}{m} = \binom{1007 + 999 + 1}{1007 + 1} = \binom{2007}{1008}. \]

### Conclusion
Thus, the coefficient of \( x^{1007} \) in the given expansion is

\[ \boxed{\binom{2007}{1008}}. \]