A bag contains three red marbles, one green one, one lavender one, two yellows, and two orange marbles. HINT [See Example 7.]
How many sets of five marbles include at most one of the yellow ones?

To solve the problem, we need to determine how many sets of five marbles can be made from the bag, with the constraint that the set includes at most one yellow marble.

### Step 1: Count the Total Marbles
- Red marbles: 3
- Green marbles: 1
- Lavender marbles: 1
- Yellow marbles: 2
- Orange marbles: 2

Total marbles in the bag: \(3 + 1 + 1 + 2 + 2 = 9\).

### Step 2: Understand the Constraints
We need to find the number of sets of 5 marbles that include at most one yellow marble. This can happen in two cases:
1. **Case 1:** The set includes 0 yellow marbles.
2. **Case 2:** The set includes exactly 1 yellow marble.

### Step 3: Calculate Each Case

#### Case 1: No Yellow Marbles
If no yellow marbles are included, we choose 5 marbles from the 7 remaining non-yellow marbles (3 red, 1 green, 1 lavender, 2 orange).

The number of ways to do this is given by the combination formula \( \binom{n}{k} \), where \( n \) is the total number of items to choose from and \( k \) is the number of items to choose.

\[
\text{Number of ways in Case 1} = \binom{7}{5} = \frac{7 \times 6}{2 \times 1} = 21
\]

#### Case 2: Exactly 1 Yellow Marble
If the set includes 1 yellow marble, we need to choose the remaining 4 marbles from the 7 non-yellow marbles.

The number of ways to do this is:

\[
\text{Number of ways in Case 2} = \binom{7}{4} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35
\]

### Step 4: Add the Results
Finally, to find the total number of sets that satisfy the condition, add the results from the two cases:

\[
\text{Total number of sets} = 21 + 35 = 56
\]

Thus, the number of sets of five marbles that include at most one of the yellow ones is **56**.

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Would you like further details or have any questions?

Here are 5 related questions:
1. How many sets of five marbles include exactly two yellow marbles?
2. What is the probability of randomly selecting a set of five marbles that includes no yellow marbles?
3. How would the solution change if the bag had 3 yellow marbles instead of 2?
4. What is the total number of different sets of five marbles that can be made from this bag?
5. How many sets of four marbles include exactly one lavender marble?

**Tip:** When solving problems involving combinations, remember to carefully identify different cases based on the conditions given.

A bag contains three red marbles, one green one, one lavender one, two yellows, and two orange marbles. How many sets of five marbles include at most one of the yellow ones?

Solve the problem of selecting sets of five marbles from a bag containing various colors, with at most one yellow marble. Learn to apply combination formulas and counting principles in this combinatorics problem suitable for Grades 9-12.

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