$$ \sum _ { k = 0 } ^ { 4 } \left( \begin{array} { l } { 4 } \ { k } \end{array} ight) ^ { x + - k } y ^ { k } $$

Question

$$ \sum _ { k = 0 } ^ { 4 } \left( \begin{array} { l } { 4 } \ { k } \end{array} ight) ^ { x + - k } y ^ { k } $$

UpStudy AI · Accepted Answer

Certainly! Let's break down and analyze the expression you've provided:

$$
\sum_{k=0}^{4} \left( \begin{array}{c} 4 \ k \end{array} \right)^{x - k} y^{k}
$$

### Understanding the Components

1. **Summation $$\sum_{k=0}^{4}$$:**
   - This indicates that the expression will be evaluated for each integer value of $$k$$ from 0 to 4 and then summed together.

2. **Binomial Coefficient $$\left( \begin{array}{c} 4 \ k \end{array} \right)$$:**
   - Also written as $$\binom{4}{k}$$, it represents the number of ways to choose $$k$$ elements from a set of 4 elements.
   - The values for $$\binom{4}{k}$$ when $$k = 0$$ to $$4$$ are:
     - $$\binom{4}{0} = 1$$
     - $$\binom{4}{1} = 4$$
     - $$\binom{4}{2} = 6$$
     - $$\binom{4}{3} = 4$$
     - $$\binom{4}{4} = 1$$

3. **Exponent $$(x - k)$$:**
   - The binomial coefficient is raised to the power of $$(x - k)$$, where $$x$$ is a variable or parameter.

4. **Term $$y^{k}$$:**
   - $$y$$ is raised to the power of $$k$$, where $$y$$ is another variable or parameter.

### Expanded Form

Let's expand the summation to see each term explicitly:

$$
\begin{align*}
\sum_{k=0}^{4} \binom{4}{k}^{x - k} y^{k} &= \binom{4}{0}^{x - 0} y^{0} + \binom{4}{1}^{x - 1} y^{1} + \binom{4}{2}^{x - 2} y^{2} + \binom{4}{3}^{x - 3} y^{3} + \binom{4}{4}^{x - 4} y^{4} \
&= 1^{x} \cdot 1 + 4^{x - 1} y + 6^{x - 2} y^{2} + 4^{x - 3} y^{3} + 1^{x - 4} y^{4} \
&= 1 + 4^{x - 1} y + 6^{x - 2} y^{2} + 4^{x - 3} y^{3} + y^{4}
\end{align*}
$$

### Simplification and Interpretation

The expression combines binomial coefficients raised to a power that depends on both $$x$$ and the summation index $$k$$, multiplied by $$y$$ raised to $$k$$. Without specific values for $$x$$ and $$y$$, we can discuss its properties:

- **Dependence on $$x$$:** The exponent $$(x - k)$$ means each term's weight changes based on $$x$$. For example, when $$x$$ increases, terms with higher binomial coefficients ($$k = 1$$ and $$k = 2$$) become more significant compared to those with lower coefficients.

- **Dependence on $$y$$:** The parameter $$y$$ scales each term by $$y^{k}$$, giving greater weight to higher $$k$$ terms as $$y$$ increases.

### Potential Applications

Such expressions might arise in contexts where weighted sums of combinatorial terms are relevant, such as:

- **Probability Theory:** Adjusting combinatorial probabilities with additional parameters.
- **Generating Functions:** Creating generating functions with modified coefficients.
- **Statistical Mechanics:** Modeling systems with varying energy states.

### Example Evaluation

Let's evaluate the sum for specific values of $$x$$ and $$y$$. Suppose $$x = 3$$ and $$y = 2$$:

$$
\begin{align*}
\sum_{k=0}^{4} \binom{4}{k}^{3 - k} 2^{k} &= 1^{3} \cdot 1 + 4^{2} \cdot 2 + 6^{1} \cdot 4 + 4^{0} \cdot 8 + 1^{-1} \cdot 16 \
&= 1 \cdot 1 + 16 \cdot 2 + 6 \cdot 4 + 1 \cdot 8 + 1 \cdot 16 \
&= 1 + 32 + 24 + 8 + 16 \
&= 81
\end{align*}
$$

### Further Assistance

If you have specific values for $$x$$ and $$y$$, or if you're looking to apply this expression in a particular context (e.g., solving an equation, modeling a scenario), please provide more details, and I'd be happy to help further!
The expression simplifies to $$1 + 4^{x - 1} y + 6^{x - 2} y^{2} + 4^{x - 3} y^{3} + y^{4}$$.

\( \sum _ { k = 0 } ^ { 4 } \left( \begin{array} { l } { 4 } \\ { k } \end{array} \right) ^ { x + - k } y ^ { k } \)

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