To determine how many sets of four marbles include all the red marbles, we can break the problem into steps.

Step 1: Identify the number of red marbles.

There are three red marbles in the bag, and we want all of them to be included in each set of four marbles.

Step 2: Determine how many more marbles are needed.

Since the set must contain four marbles and we already have three red marbles, we need to select one additional marble from the other non-red marbles.

Step 3: Count the remaining non-red marbles.

• Green marbles: 5
• Lavender marble: 1
• Yellow marbles: 6
• Orange marbles: 3

The total number of non-red marbles is: $5 + 1 + 6 + 3 = 15$

Step 4: Calculate the number of ways to select one marble from the 15 non-red marbles.

The number of ways to choose 1 marble from 15 is given by the combination formula: $\binom{15}{1} = 15$

Final Answer:

There are 15 different sets of four marbles that include all three red marbles.

Would you like further details or have any other questions?

Here are 5 related questions:

1. How would the number change if we wanted a set of five marbles?
2. What is the total number of possible sets of four marbles without restrictions?
3. How many sets of four marbles include exactly two red marbles?
4. What is the probability of randomly selecting a set of four marbles that includes all three red ones?
5. How many sets of four marbles include none of the red marbles?

Tip: In probability problems, identifying and organizing the groups of interest can help simplify the solution steps.