Let S = {∞ · a, ∞ · b, ∞ · c}, answer the following:
 
(d) How many 10 - combinations of S contain at least two a’s and at least 2 b’s?

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.

Let S = {∞ · a, ∞ · b, ∞ · c}, answer the following:
(d) How many 10-combinations of S contain at least two a’s and at least 2 b’s?

Explore how to solve a combinatorics problem involving 10-combinations of the set {∞ · a, ∞ · b, ∞ · c}, where each element can appear infinitely many times. This problem applies the stars and bars method to find the number of combinations that contain at least two a’s and two b’s. Suitable for Grades 10-12 students studying combinatorics.

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