Let's address both parts step by step:

Part (a): $x_1 + x_2 + x_3 + x_4 + x_5 = 25$, where $x_i > 0$

To solve this, we use the "stars and bars" method, which is commonly used to count the number of ways to distribute indistinguishable objects (stars) into distinct groups (bars).

Since $x_i > 0$, we can first transform the variables by setting $y_i = x_i - 1$, where $y_i \geq 0$. This way, the equation becomes:

$y_1 + y_2 + y_3 + y_4 + y_5 = 25 - 5 = 20$

Now, we need to count how many non-negative integer solutions exist for this equation. The number of non-negative integer solutions to an equation of the form:

$y_1 + y_2 + \cdots + y_k = n$

is given by the binomial coefficient $\binom{n+k-1}{k-1}$. In our case, $n = 20$ and $k = 5$, so the number of solutions is:

$\binom{20+5-1}{5-1} = \binom{24}{4}$

Now, let's calculate $\binom{24}{4}$:

$\binom{24}{4} = \frac{24 \times 23 \times 22 \times 21}{4 \times 3 \times 2 \times 1} = 10626$

Thus, there are 10,626 different integer value solutions for part (a).

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Part (b): $x_1 + x_2 + x_3 = 15$, where $x_i \geq 0$

Here, we again use the "stars and bars" method, but this time with non-negative integers $x_i \geq 0$.

The number of non-negative integer solutions to this equation is given by the binomial coefficient $\binom{n+k-1}{k-1}$, where $n = 15$ and $k = 3$.

$\binom{15+3-1}{3-1} = \binom{17}{2}$

Now, let's calculate $\binom{17}{2}$:

$\binom{17}{2} = \frac{17 \times 16}{2 \times 1} = 136$

Thus, there are 136 different integer value solutions for part (b).

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Summary:

• (a) There are 10,626 solutions where $x_1 + x_2 + x_3 + x_4 + x_5 = 25$ with $x_i > 0$.
• (b) There are 136 solutions where $x_1 + x_2 + x_3 = 15$ with $x_i \geq 0$.

Would you like more details or clarifications?

Here are 5 related questions:

1. How does the stars and bars method work in combinatorics?
2. Can the stars and bars method handle equations with more complex constraints, such as $x_i \leq n$?
3. How would the solution change if there were upper limits on $x_i$?
4. What happens if $x_i$ are required to be even integers?
5. How would we solve the equations if $x_i$ must be prime numbers?

Tip: The stars and bars method is powerful for partitioning problems. Always check if the constraints on variables are strict (e.g., $x_i > 0$) or flexible (e.g., $x_i \geq 0$).