The number of outcomes​ N, in which a die is rolled 8 times in a row and the sum of the numbers rolled is​ observed, is?

When rolling a die 8 times in a row, and observing the sum of the numbers rolled, the number of possible outcomes is calculated based on the fact that each roll can result in one of 6 possible numbers (1 through 6). However, we are interested in the number of distinct possible sums that can result from these rolls.

### Understanding the Problem:
- Each die roll can result in a number between 1 and 6.
- We roll the die 8 times, so we are adding 8 numbers, each between 1 and 6.

### Step-by-Step Solution:
1. **Minimum Sum**: The minimum sum occurs when each roll results in the lowest possible number, which is 1. Thus, the minimum sum is:
   \[
   1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 8
   \]

2. **Maximum Sum**: The maximum sum occurs when each roll results in the highest possible number, which is 6. Thus, the maximum sum is:
   \[
   6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 = 48
   \]

3. **Number of Outcomes \( N \)**: Each roll of the die is independent, and there are 6 possible outcomes for each roll. Since the die is rolled 8 times, the total number of possible outcomes is:
   \[
   N = 6^8 = 1679616
   \]

This means there are 1,679,616 different possible sequences of rolls, though some of these sequences will result in the same sum.

However, calculating the exact number of distinct possible sums that can occur is a more complex problem typically addressed with generating functions or dynamic programming, and it requires a combinatorial analysis.

### Conclusion:
The total number of possible outcomes \( N \) where a die is rolled 8 times is **1,679,616**. But this includes all possible sequences, not just distinct sums.

Would you like to explore further on distinct sums, or do you have other questions related to this topic?

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**Follow-up Questions:**
1. How would the number of outcomes change if the die is rolled 10 times instead of 8?
2. What is the probability of obtaining a sum of 24 when rolling a die 8 times?
3. How can generating functions be used to find the number of distinct sums?
4. What is the expected sum when a die is rolled 8 times?
5. How does the probability distribution of the sum behave as the number of rolls increases?

**Tip:** Understanding the concept of **independent events** is crucial when dealing with probabilities and outcomes in scenarios involving repeated trials, such as rolling a die multiple times.

Discover how to calculate the number of possible outcomes when rolling a die 8 times in a row. This problem involves key concepts from probability and combinatorics, including the rule of product and basic counting principles. Suitable for students in Grades 9-12.

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