To find all integer solutions for $a + b + c \leq 25$, where $a, b, c$ are integers, we can approach the problem as follows:

1. Basic Setup: $a + b + c \leq 25 \quad \text{where} \quad a, b, c \in \mathbb{Z}$ This inequality implies that the sum of three integers is at most 25.

2. Generalization: The solutions are triples $(a, b, c)$ where the sum of the three integers is less than or equal to 25. The individual integers could vary from negative to positive values, as long as the total sum doesn't exceed 25.

3. Simplified Boundaries: Assume a lower bound for simplicity, such as $a, b, c \geq 0$ (non-negative integers). In this case, we are looking for non-negative integer solutions to: $a + b + c \leq 25$

Step-by-Step Process

1. Counting non-negative integer solutions: For non-negative integers $a, b, c$, the number of solutions to $a + b + c \leq 25$ can be counted using stars and bars (a combinatorial method). This transforms the inequality into a combination problem. The total number of solutions is equivalent to counting the number of ways to distribute 25 units (since the maximum value of the sum is 25) into 3 bins (corresponding to $a$, $b$, and $c$).

2. Stars and Bars Formula: The formula for the number of solutions to $x_1 + x_2 + \dots + x_k = n$ for non-negative integers is given by: $\text{Number of solutions} = \binom{n+k-1}{k-1}$ Here $n = 25$ and $k = 3$ (since we have 3 variables). Therefore, the number of non-negative integer solutions to $a + b + c \leq 25$ is: $\sum_{n=0}^{25} \binom{n+2}{2}$

However, if we also consider negative values for $a$, $b$, and $c$, the problem becomes more complex. Would you like me to continue with the combinatorial approach or extend it to allow negative values for $a$, $b$, and $c$?

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Follow-up Questions:

1. How would the solutions change if we restricted $a, b, c \geq 0$?
2. Would you like to explore solutions where $a, b, c$ can be negative?
3. How would you count the number of solutions for specific cases, such as $a + b + c = 25$?
4. What role does combinatorics play in solving this kind of inequality?
5. Would you like to explore how stars and bars apply to more complex inequalities?

Tip:

For problems with multiple variables under a sum constraint, the stars and bars method is a powerful tool for counting the number of non-negative integer solutions.