A bag contains three red marbles, five green ones one lavender one three yellows and three orange marbles. How many sets of four marbles include all the red ones.

To solve this problem, we need to determine how many sets of four marbles from the bag include all three red marbles.

### Total marbles in the bag:
- Red marbles: 3
- Green marbles: 5
- Lavender marbles: 1
- Yellow marbles: 3
- Orange marbles: 3

This gives a total of:
\[
3 \,(\text{red}) + 5 \,(\text{green}) + 1 \,(\text{lavender}) + 3 \,(\text{yellow}) + 3 \,(\text{orange}) = 15 \, \text{marbles}.
\]

### Step-by-step solution:

1. We need a set of **4 marbles** that includes **all 3 red marbles**. That means 3 marbles are already chosen (the red ones).
2. We are left with \( 15 - 3 = 12 \) marbles (non-red ones) to choose the **fourth** marble.

### Conclusion:
We need to choose 1 marble from the remaining 12 non-red marbles. This can be done in:
\[
\binom{12}{1} = 12 \, \text{ways}.
\]

Thus, there are **12 sets** of 4 marbles that include all the red marbles.

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

Here are 5 related questions:
1. How many total sets of 4 marbles can be chosen from the bag?
2. What if we want sets with at least 2 red marbles?
3. How many sets include exactly 1 red marble?
4. How many sets include no red marbles at all?
5. If we add another color to the bag, how does that change the combinations?

**Tip**: The binomial coefficient \(\binom{n}{k}\) gives the number of ways to choose \(k\) objects from \(n\) without regard to the order.

A bag contains three red marbles, five green ones, one lavender one, three yellows, and three orange marbles. How many sets of four marbles include all the red ones?

Explore how to calculate the number of sets of four marbles that include all red marbles from a collection of 15 marbles, using combinatorial methods. This step-by-step solution demonstrates key concepts such as the binomial coefficient and combinatorics, suitable for Grades 8-10.

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