For the given decision algorithm, find how many outcomes are possible. HINT [See Example 1.]
Step 1:		Step 2:
Alternative 1: 1 outcome		    Alternative 1: 3 outcomes
Alternative 2: 4 outcomes     		    Alternative 2: 5 outcomes
Alternative 3: 1 outcome

outcomes

To find the total number of possible outcomes from this decision algorithm, we'll apply the multiplication rule of counting. This rule states that if you have multiple steps or stages in a process, the total number of possible outcomes is the product of the number of choices (or outcomes) available at each step.

### Step-by-step breakdown:

#### Step 1:
- There are three alternatives in Step 1, each with a specific number of outcomes:
  1. **Alternative 1**: 1 outcome
  2. **Alternative 2**: 4 outcomes
  3. **Alternative 3**: 1 outcome

#### Step 2:
- Each alternative in Step 1 leads to a choice in Step 2. For each alternative, there are the following outcomes:
  1. **Alternative 1 of Step 2**: 3 outcomes
  2. **Alternative 2 of Step 2**: 5 outcomes

To find the total number of outcomes, we multiply the number of outcomes for each step:

1. **Alternative 1**:  
   - Step 1: 1 outcome  
   - Step 2: 3 outcomes  
   Total outcomes for Alternative 1: \(1 \times 3 = 3\)

2. **Alternative 2**:  
   - Step 1: 4 outcomes  
   - Step 2: 5 outcomes  
   Total outcomes for Alternative 2: \(4 \times 5 = 20\)

3. **Alternative 3**:  
   - Step 1: 1 outcome  
   - Step 2: 3 outcomes (same as Alternative 1)  
   Total outcomes for Alternative 3: \(1 \times 3 = 3\)

### Final total:
Now, sum the outcomes for all alternatives:
\[
3 \ (\text{Alternative 1}) + 20 \ (\text{Alternative 2}) + 3 \ (\text{Alternative 3}) = 26
\]

Thus, the total number of possible outcomes is **26**.

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Would you like more details or have any questions about this?

Here are 5 related questions to further expand on this:
1. What if there were a fourth alternative in Step 1 with 2 outcomes?
2. How would the process change if the number of alternatives in Step 2 increased?
3. Can you explain how the multiplication rule applies in probability theory?
4. What are the possible outcomes if Step 2 had only one alternative?
5. How does the decision tree approach help visualize this problem?

**Tip**: When solving counting problems, carefully check how each stage interacts with the next to avoid missing any combinations!

For the given decision algorithm, find how many outcomes are possible. HINT [See Example 1.] Step 1: Step 2: Alternative 1: 1 outcome Alternative 1: 3 outcomes Alternative 2: 4 outcomes Alternative 2: 5 outcomes Alternative 3: 1 outcome

Explore how to calculate the total number of outcomes in a decision algorithm with multiple steps, applying the multiplication rule of counting. This problem breaks down each step with alternatives and explains how to find the final total. Suitable for students in Grades 9-12.

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