The coefficient of `x^(9)` in the expansion of `E = (1 + x)^(9) + (1 + x)^(10) + ... + (1 + x)^(100)` is

Options

A. `""^(101)C_(9)`
B. `""^(101)C_(10)`
C. `""^(100)C_(9)`
D. `""^(100)C_(10)`

Answer

To find the coefficient of \( x^9 \) in the expansion of

\[ E = (1 + x)^{9} + (1 + x)^{10} + \ldots + (1 + x)^{100}, \]

we can follow these steps:

### Step 1: Identify the Coefficient of \( x^9 \) in Each Term
The coefficient of \( x^9 \) in the expansion of \( (1 + x)^n \) is given by \( \binom{n}{9} \). Therefore, we need to find the coefficients for each term from \( n = 9 \) to \( n = 100 \).

### Step 2: Write the Sum of Coefficients
We can express the total coefficient of \( x^9 \) in \( E \) as:

\[ \text{Total Coefficient} = \sum_{n=9}^{100} \binom{n}{9}. \]

### Step 3: Use the Hockey Stick Identity
The Hockey Stick Identity states that:

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

In our case, we can apply this identity with \( r = 9 \) and \( n = 100 \):

\[ \sum_{n=9}^{100} \binom{n}{9} = \binom{100 + 1}{9 + 1} = \binom{101}{10}. \]

### Step 4: Calculate the Coefficient
Now, we need to compute \( \binom{101}{10} \):

\[ \binom{101}{10} = \frac{101!}{10!(101-10)!} = \frac{101!}{10! \cdot 91!}. \]

This gives us the coefficient of \( x^9 \) in the expansion of \( E \).

### Final Answer
Thus, the coefficient of \( x^9 \) in the expansion of \( E \) is:

\[ \boxed{\binom{101}{10}}. \]