The coefficient of x¹⁰⁰⁷ in the expansion of

(1 + x)²⁰⁰⁶ + x(1 + x)²⁰⁰⁵ + x²(1 + x)²⁰⁰⁴ + x³(1 + x)²⁰⁰³ + … + x²⁰⁰⁶ is

(1) ²⁰⁰⁶C₁₀₀₇

(2) ²⁰⁰⁶C₁₀₀₆

(3) ²⁰⁰⁷C₁₀₀₆

(4) ²⁰⁰⁷C₁₀₀₇

To find the coefficient of \( x^{1007} \) in the given expression, we first need to analyze the series: \( (1 + x)^{2006} + x(1 + x)^{2005} + x^2(1 + x)^{2004} + \ldots + x^{2006} \). This can be rewritten as a summation, which helps us understand the contribution of each term in the series.

Breaking Down the Expression

The expression can be expressed as:

• Term 1: \( (1 + x)^{2006} \)
• Term 2: \( x(1 + x)^{2005} \)
• Term 3: \( x^2(1 + x)^{2004} \)
• ...
• Term 2007: \( x^{2006} \)

Each term contributes to the overall expansion, and we can see that the \( k \)-th term is \( x^{k}(1 + x)^{2006-k} \) for \( k = 0, 1, 2, \ldots, 2006 \).

Finding the Coefficient of \( x^{1007} \)

To find the coefficient of \( x^{1007} \), we need to consider how each term contributes to \( x^{1007} \). The \( k \)-th term contributes \( x^{k} \) multiplied by the expansion of \( (1 + x)^{2006-k} \). Thus, we need to find the coefficient of \( x^{1007-k} \) in \( (1 + x)^{2006-k} \).

Using the binomial theorem, the coefficient of \( x^m \) in \( (1 + x)^n \) is given by \( \binom{n}{m} \). Therefore, the coefficient of \( x^{1007-k} \) in \( (1 + x)^{2006-k} \) is \( \binom{2006-k}{1007-k} \).

Summing Contributions

Now, we need to sum the contributions from all terms where \( k \) ranges from 0 to 1007 (since \( k \) cannot exceed 1007 for \( x^{1007} \) to be formed). The total coefficient of \( x^{1007} \) is:

\[ \sum_{k=0}^{1007} \binom{2006-k}{1007-k} \]

This can be simplified by changing the index of summation. Let \( j = 1007 - k \), which means \( k = 1007 - j \). The limits change accordingly: when \( k = 0 \), \( j = 1007 \), and when \( k = 1007 \), \( j = 0 \). Thus, we can rewrite the sum as:

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

Using the Hockey Stick Identity

This sum can be evaluated using the Hockey Stick Identity in combinatorics, which states:

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

Applying this identity here gives us:

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

Final Result

Thus, the coefficient of \( x^{1007} \) in the original expression is \( \binom{2007}{1007} \). Therefore, the correct answer is:

(4) \( 2007 C 1007 \)