The sum of the series
`""^(4)C_(0) + ""^(5)C_(1) x + ""^(6)C_(2) x^(2)+""^(7)C_(3)x^(3) +... ` to `infty`, is

Options

A. `(1)/((1 - X)^(5)`
B. `(1)/( 1 - X)^(5)`
C. `(1 + X)^(-5)`
D. none of these

Explanation

To find the sum of the series

\[ \sum_{n=0}^{\infty} \binom{n+4}{4} x^n \]

we can use the binomial theorem and properties of generating functions. The series can be rewritten using the binomial coefficient notation.

### Step 1: Identify the General Term
The general term of the series is given by:

\[ \binom{n+4}{4} x^n \]

### Step 2: Recognize the Binomial Series
From the binomial theorem, we know that:

\[ (1 - x)^{-k} = \sum_{n=0}^{\infty} \binom{n+k-1}{k-1} x^n \]

For our case, we can set \(k = 5\). Thus, we have:

\[ (1 - x)^{-5} = \sum_{n=0}^{\infty} \binom{n+4}{4} x^n \]

### Step 3: Write the Sum of the Series
Now, we can express the sum of the series as:

\[ \sum_{n=0}^{\infty} \binom{n+4}{4} x^n = (1 - x)^{-5} \]

### Step 4: Final Expression
Thus, the sum of the series is:

\[ \frac{1}{(1 - x)^5} \]

### Conclusion
The final answer for the sum of the series

\[ \sum_{n=0}^{\infty} \binom{n+4}{4} x^n = \frac{1}{(1 - x)^5} \]