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The problem involves a test with 10 multiple-choice questions. Each correct answer is worth 5 points, each unanswered question is worth 1 point, and each incorrect answer is worth 0 points. The task is to determine the maximum possible average score if \( N \) is equal to the number of distinct integral scores possible between 30 and 50, inclusive.

### Step-by-Step Solution

1. **Determine Possible Scores:**  
   Since each correct answer is worth 5 points, a participant can score between \( 0 \times 5 = 0 \) (for all incorrect answers) and \( 10 \times 5 = 50 \) (for all correct answers).
   
   Let \( x \) be the number of correct answers, \( y \) the number of unanswered questions, and \( z \) the number of incorrect answers. Then:
   - \( x + y + z = 10 \)
   - Total score \( = 5x + 1y + 0z = 5x + y \)

2. **Calculate Scores Between 30 and 50:**  
   We need to find combinations of \( x \) and \( y \) (with \( z = 10 - x - y \)) such that \( 30 \leq 5x + y \leq 50 \).
   
   - **For \( x = 6 \):**  
     \( 5 \times 6 = 30 \), so \( y \) can be \( 0 \), resulting in a score of 30.

   - **For \( x = 7 \):**  
     \( 5 \times 7 = 35 \), and \( y \) can vary between 0 and 5 (to keep total \( x + y + z = 10 \)). This gives scores from 35 to 40.

   - **For \( x = 8 \):**  
     \( 5 \times 8 = 40 \), and \( y \) can vary between 0 and 2. This gives scores from 40 to 42.

   - **For \( x = 9 \):**  
     \( 5 \times 9 = 45 \), and \( y \) can be 0 or 1, giving scores of 45 and 46.

   - **For \( x = 10 \):**  
     \( 5 \times 10 = 50 \), and \( y = 0\) (all answers are correct). The score is 50.

   Thus, the possible distinct scores between 30 and 50 are:  
   \( 30, 35, 36, 37, 38, 39, 40, 41, 42, 45, 46, 50 \).

3. **Count Possible Scores (N):**  
   The distinct scores between 30 and 50 are 12.

4. **Maximum Possible Average Score:**  
   The highest score among these distinct scores is 50, so the maximum possible average score is 50 if all friends scored the maximum possible score.

### Final Answer
The maximum possible average score is:
**Option 4) 46**

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Would you like a further breakdown of this solution, or do you have any questions?

Here are some related questions for further practice:
1. If the range was extended to scores between 20 and 50, what would be the new maximum average score?
2. What would be the minimum possible average score for the given constraints?
3. How would the answer change if each correct answer was worth 6 points instead of 5?
4. If only half of the friends scored between 30 and 50, what would the new average be?
5. Can you calculate the average score if each friend scored randomly between 30 and 50?

**Tip:** When dealing with constraints in word problems, try breaking them down into manageable parts to isolate specific values.

N friends gave a test and obtained a distinct integral score. The test was multiple choice with 10 questions on it. Each question answered correctly is worth 5 points, each unanswered question is worth 1 point, and each question answered incorrectly is worth 0 points. If N is equal to the number of integral scores not possible between 30 and 50, both inclusive, then what is the maximum possible average score obtained by these friends?

Explore how to determine the maximum possible average score for N friends in a multiple-choice test. Each correct answer earns 5 points, unanswered questions are worth 1 point, and wrong answers score 0 points. Find out how to calculate scores within a range of 30 to 50, and determine the maximum possible average for this setup.

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