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: 2 outcomes
    Alternative 2: 5 outcomes     		    Alternative 2: 5 outcomes
    Alternative 3: 1 outcome


To determine how many outcomes are possible for the given decision algorithm, we need to calculate the total number of possible outcomes by following each step in the algorithm.

### Step 1:
- **Alternative 1**: 1 outcome
- **Alternative 2**: 5 outcomes
- **Alternative 3**: 1 outcome

In Step 1, there are 3 alternatives. Therefore, the total number of outcomes for Step 1 is:
\[
1 + 5 + 1 = 7 \text{ outcomes}
\]

### Step 2:
- **Alternative 1**: 2 outcomes
- **Alternative 2**: 5 outcomes

In Step 2, there are 2 alternatives. The total number of outcomes for Step 2 is:
\[
2 + 5 = 7 \text{ outcomes}
\]

### Total Possible Outcomes:
Since each step is independent of the other, we multiply the total outcomes from Step 1 and Step 2 to get the total number of possible outcomes for the entire decision algorithm. Thus:
\[
7 \times 7 = 49 \text{ outcomes}
\]

### Final Answer:
There are **49 possible outcomes** for the given decision algorithm.

Would you like further details on how this is computed or any other clarification?

### Related Questions:
1. How would the total outcomes change if there were additional alternatives in Step 1 or Step 2?
2. Can you describe how independent decision-making steps affect the total outcomes?
3. How would the calculation change if the alternatives were dependent on each other?
4. What would happen if a third step was introduced with different alternatives?
5. How do different algorithms compare in terms of outcome complexity?

### Tip:
Always check if decision steps are independent or dependent, as this affects how outcomes are calculated.

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: 2 outcomes Alternative 2: 5 outcomes Alternative 2: 5 outcomes Alternative 3: 1 outcome

Learn how to calculate the total number of outcomes for a decision algorithm with multiple alternatives. This example walks through the calculation using the multiplication principle of counting, suitable for high school students studying combinatorics and decision-making algorithms.

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