What is the arithmetic mean (average) of the multiset of numbers 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6 and 7, 7, 7, 7, 7, 7, 7?

Introduction / Context:
This question tests basic statistics, specifically the computation of an arithmetic mean when numbers are repeated. Rather than listing every number individually, we can take advantage of the fact that each value occurs a known number of times. The arithmetic mean is a measure of central tendency and is computed by dividing the total sum of all data values by the total number of data values.

Given Data / Assumptions:
- The multiset of numbers is: one 1, two 2s, three 3s, four 4s, five 5s, six 6s, and seven 7s.
- All numbers are to be included when calculating the mean, with their repetitions counted appropriately.
- The arithmetic mean is defined as (sum of all observations) / (number of observations).

Concept / Approach:
Instead of writing out each number explicitly, we can use the frequencies directly. For every distinct value v that appears f times, it contributes v * f to the sum and f to the count. We sum these contributions for all distinct values and then divide the total sum by the total count. This is efficient and helps avoid arithmetic mistakes when there are many repetitions.

Step-by-Step Solution:
Step 1: Count how many times each number appears: 1 appears once, 2 appears twice, 3 appears three times, 4 appears four times, 5 appears five times, 6 appears six times, and 7 appears seven times. Step 2: Compute the total number of observations: 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28. Step 3: Compute the total sum using value × frequency:
1 × 1 = 1,
2 × 2 = 4,
3 × 3 = 9,
4 × 4 = 16,
5 × 5 = 25,
6 × 6 = 36,
7 × 7 = 49. Step 4: Add these partial sums: 1 + 4 + 9 + 16 + 25 + 36 + 49 = 140. Step 5: Use the mean formula: mean = total sum / total count = 140 / 28. Step 6: Simplify 140 / 28 = 5.

Verification / Alternative check:
An alternative way is to notice that the pattern is symmetric from 1 to 7, with increasing frequencies, but computation via value times frequency is already simple and direct. Dividing both numerator and denominator by 7 shows that 140 / 28 = (20 × 7) / (4 × 7) = 20 / 4 = 5. Both methods confirm that the arithmetic mean is 5.

Why Other Options Are Wrong:
Option 6: This is larger than the majority of the data points and does not match the computed average.
Option 7: This would be the maximum value, but the presence of many smaller numbers pulls the average down.
Option 9: Far exceeds any individual data point and cannot be the mean.
Option None of these: Since 5 is one of the options and matches the correct mean, this option is incorrect.

Common Pitfalls:
Students sometimes forget to count repetitions correctly or mistakenly divide by the number of distinct values instead of the total number of observations. Another common error is arithmetic slips when summing many terms. Using the value × frequency method systematically, and double-checking sums, reduces the risk of such mistakes.

Final Answer:
The arithmetic mean of the given multiset of numbers is 5.