The equation in the image is given as:

$abc + ab + bc + ca + a + b + c = 2024$

We are asked to find the sum of all possible values of $c$, where $a$, $b$, and $c$ are positive integers such that $a < b < c$.

To approach this, the expression can be rewritten as:

$abc + ab + bc + ca + a + b + c = 2024$

This is a symmetric equation in $a$, $b$, and $c$, meaning the variables can be swapped without changing the equation. However, since $a < b < c$, we can focus on solutions with $a < b < c$.

Steps to solve:

1. Trial and Error Strategy: Since this is a Diophantine equation with positive integer solutions, we can manually substitute values for $a$ and $b$, solve for $c$, and check which combinations satisfy the equation.

2. Reduce and Factorize: The equation resembles a polynomial form, but rewriting it directly using algebraic manipulation seems impractical because of the complex interactions between $a$, $b$, and $c$. Therefore, a systematic search will help us determine the possible values of $c$.

Let me proceed with this method to find the possible values of $c$ and their sum.The possible values of $c$ that satisfy the equation are $26$, $44$, and $74$. The sum of all these different values of $c$ is:

$26 + 44 + 74 = 144$

Would you like more details on how this was computed or explanations on any specific step?

Here are 5 related questions you might find helpful:

1. What are some alternative methods to solving this type of Diophantine equation?
2. How can we prove that there are no other possible values of $c$?
3. How would this problem change if we allowed non-integer solutions for $a$, $b$, and $c$?
4. What role do Vieta’s formulas play in simplifying symmetric polynomial equations?
5. How can constraints like $a < b < c$ be exploited in mathematical problem-solving?

Tip: When solving Diophantine equations, simplifying the equation using common algebraic techniques or inequalities can often narrow down the search space for potential solutions!