The bag contains 3 red, 2 green, 1 purple, 2 yellow, and 2 orange marbles.Find the number of sets of picking up 4 marbles at a time that contain all the red marbles available.

To find the number of sets of picking up 4 marbles that contain all the 3 red marbles, we need to first consider that 3 of the 4 marbles are already determined (they must be red). This leaves us with the task of choosing 1 additional marble from the remaining non-red marbles.

Here's the breakdown:

1. **Number of red marbles**: 3 (which must be included in the set).
2. **Total number of non-red marbles**: \(10 - 3 = 7\).

These 7 non-red marbles consist of:
- 2 green
- 1 purple
- 2 yellow
- 2 orange

Now, we need to choose 1 marble from these 7 non-red marbles.

The number of ways to choose 1 marble from 7 is calculated as follows:

\[
C(7, 1) = \frac{7!}{1!(7-1)!} = 7
\]

Thus, the number of sets of picking up 4 marbles that contain all 3 red marbles is **7**.

Would you like more details on this calculation or have any other questions?

Here are 5 related questions:
1. How would the result change if there were only 2 red marbles?
2. What if you were asked to pick 5 marbles containing all the red ones?
3. How many sets of 4 marbles can be picked if the set must include exactly 2 red marbles?
4. What is the probability of picking a set of 4 marbles that includes all the red ones?
5. How does the number of sets change if more colors or marbles were added?

**Tip:** When dealing with combinations, it helps to separate the problem into smaller steps—first fixing the required items, then considering the choices left for the remaining items.

Learn how to calculate the number of sets of 4 marbles containing all the red marbles from a bag with 3 red, 2 green, 1 purple, 2 yellow, and 2 orange marbles. Understand the application of combinations in this problem, where 3 of the 4 marbles are predetermined (red), and one additional marble needs to be chosen from the remaining non-red marbles.

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