A bag contains three red marbles, four green ones, one lavender one, three yellows, and two orange marbles. HINT [See Example 7.]
How many sets of five marbles include either the lavender one or exactly one yellow one but not both colors?

We are tasked with finding how many sets of five marbles include either the lavender marble or exactly one yellow marble, but **not** both.

### Step-by-step solution:

We start by calculating the total number of marbles:
- 3 red
- 4 green
- 1 lavender
- 3 yellow
- 2 orange

This gives a total of:
\[
3 + 4 + 1 + 3 + 2 = 13 \text{ marbles.}
\]

We are looking for sets of five marbles that include either:
- **The lavender marble**, or
- **Exactly one yellow marble**, but **not both** colors (lavender and exactly one yellow).

### Case 1: The lavender marble is included, but no yellow marble.
- We have 1 lavender marble, so 4 more marbles must be selected from the non-yellow marbles. 
- The non-yellow marbles consist of:
  - 3 red
  - 4 green
  - 2 orange
  So, there are 9 non-yellow marbles to choose from.
  
The number of ways to select 4 marbles from these 9 is:
\[
\binom{9}{4} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126.
\]

### Case 2: Exactly one yellow marble is included, but no lavender marble.
- We must choose 1 yellow marble, and the remaining 4 marbles must come from the non-lavender, non-yellow marbles.
- The non-lavender, non-yellow marbles consist of:
  - 3 red
  - 4 green
  - 2 orange
  So, there are 9 such marbles to choose from.

The number of ways to select 4 marbles from these 9 is again:
\[
\binom{9}{4} = 126.
\]

### Total number of sets:
Since the two cases (lavender included or exactly one yellow included) are mutually exclusive, we add the results from both cases:
\[
126 + 126 = 252.
\]

Thus, the total number of sets of five marbles that include either the lavender marble or exactly one yellow but not both is **252**.

Would you like any further clarification or a deeper explanation?

Here are 5 related questions to explore:

1. What if the set included at most one yellow marble but still had the same conditions?
2. How would the result change if there were two lavender marbles instead of one?
3. How many sets of five marbles include at least two green marbles?
4. What if we wanted to exclude all red marbles in the selection of five?
5. How many ways can we select exactly three green marbles from the set?

**Tip:** When calculating combinations, always check if the selection categories are mutually exclusive or overlap. This helps avoid counting errors.

A bag contains three red marbles, four green ones, one lavender one, three yellows, and two orange marbles. How many sets of five marbles include either the lavender one or exactly one yellow one but not both colors?

Explore how to calculate the number of sets of five marbles that include either the lavender marble or exactly one yellow but not both, using combinatorics. This problem applies the combination formula and counting principles, suitable for students in Grades 10-12.

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