We are given a set $S = \{\infty \cdot a, \infty \cdot b, \infty \cdot c\}$, where the elements $a$, $b$, and $c$ have infinite repetition. The problem asks for how many 10-combinations of $S$ contain at least 2 $a$'s and at least 2 $b$'s.

Step 1: Understanding the Setup

A 10-combination of $S$ is a combination where the sum of the occurrences of $a$'s, $b$'s, and $c$'s adds up to 10. In mathematical terms, this is finding the number of solutions to: $x_a + x_b + x_c = 10$ where $x_a$, $x_b$, and $x_c$ are non-negative integers representing how many times $a$, $b$, and $c$ appear in the combination.

Step 2: Enforcing the Conditions

We are asked to find combinations that contain at least two $a$'s and at least two $b$'s. To enforce this condition, let’s introduce new variables: $y_a = x_a - 2, \quad y_b = x_b - 2$ Thus, $y_a \geq 0$ and $y_b \geq 0$, and now the equation becomes: $(y_a + 2) + (y_b + 2) + x_c = 10$ Simplifying, we get: $y_a + y_b + x_c = 6$ Now we need to count the number of non-negative integer solutions to this equation.

Step 3: Applying the Stars and Bars Method

The equation $y_a + y_b + x_c = 6$ is a standard stars and bars problem, where we count the number of non-negative integer solutions to an equation of the form $y_a + y_b + x_c = n$.

The formula for the number of non-negative integer solutions to $y_1 + y_2 + \dots + y_k = n$ is given by: $\binom{n+k-1}{k-1}$ In our case, $n = 6$ and $k = 3$ (since we have 3 variables: $y_a$, $y_b$, and $x_c$). Therefore, the number of solutions is: $\binom{6+3-1}{3-1} = \binom{8}{2}$ Now, calculate: $\binom{8}{2} = \frac{8 \times 7}{2} = 28$

Step 4: Conclusion

There are 28 10-combinations of $S$ that contain at least two $a$'s and at least two $b$'s.

Would you like further details on any step, or have questions?

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Here are 5 related questions to explore:

1. How many 10-combinations contain exactly 3 $a$'s and 4 $b$'s?
2. What is the total number of 10-combinations of $S$ without any restrictions?
3. How many 10-combinations contain at most 1 $c$?
4. What is the number of 12-combinations that contain at least 4 $a$'s and 3 $b$'s?
5. How many 10-combinations contain no $a$'s and no $c$'s?

Tip: The stars and bars method is a powerful combinatorial technique to solve problems involving distributing indistinguishable objects into distinguishable boxes.