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In this problem, Pablo is randomly selecting **three marbles** from a bag containing **nine marbles** in total: 4 red, 3 green, and 2 yellow. Let's break it down.

### 1. **How many outcomes are there in the sample space?**
The total number of ways to choose 3 marbles from 9 can be calculated using the **combination formula**:

\[
\text{Combinations} = \binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

Where:
- \(n\) is the total number of marbles (9),
- \(k\) is the number of marbles chosen (3).

So, we calculate:

\[
\binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1}
\]

### 2. **How many outcomes in the event that none of the marbles he picks are red?**
If none of the marbles picked are red, then Pablo is choosing from only the **green and yellow marbles**. There are:
- 3 green marbles,
- 2 yellow marbles.

This gives us a total of 5 marbles, and we need to choose 3 from these 5. Using the combination formula again:

\[
\binom{5}{3} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1}
\]

Let me compute these values for you.Here are the answers to the two parts of the question:

1. **Total outcomes in the sample space** (selecting 3 marbles from 9): \( 84 \) outcomes.
2. **Outcomes where none of the marbles are red** (selecting 3 marbles from 5 green and yellow): \( 10 \) outcomes.

Would you like further explanation or help with any additional questions?

Here are 5 related questions to explore this concept further:
1. How many outcomes involve selecting exactly 2 red marbles?
2. What is the probability of selecting 3 red marbles?
3. If the marbles were placed back in the bag after each pick, how would the outcome change?
4. How many ways are there to select 1 green, 1 yellow, and 1 red marble?
5. What is the probability of selecting at least 1 yellow marble?

**Tip:** Remember, combinations are used when the order of selection does not matter.

Pablo randomly picks three marbles from a bag of nine marbles (four red, three green, and two yellow ones). How many outcomes are there in the sample space? How many outcomes in the event that none of the marbles he picks are red?

Explore how to calculate the number of outcomes when selecting 3 marbles from a set of 9 using combinations. Learn to solve questions involving outcomes where none of the marbles are red, utilizing the combination formula. Suitable for students in Grades 8-10.

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