Fill in the blanks such that the median of the collection is 13: 5, 21, 14, ___, ___, ___. How many possibilities exist if only counting numbers are allowed?
Given: Collection: 5, 21, 14, ___, ___, ___ (6 values total)
Step 1: Sort the known values: 5, 14, 21
Step 2: With 6 total values, median = average of 3rd and 4th values when sorted
Step 3: For median = 13: (3rd + 4th)/2 = 13 ∴ 3rd + 4th = 26
Step 4: The three blanks must be counting numbers that allow this condition
Answer: Multiple possibilities exist depending on the values chosen.
Explanation
The chapter explains that for an even number of values, the median is the average of the two middle values. With 6 values sorted, positions 3 and 4 determine the median. For median 13, these two values must sum to 26.
Solution Steps
Step 1: Sort the known values: 5, 14, 21
Step 2: With 6 total values, median = average of 3rd and 4th values
Step 3: For median = 13, (3rd + 4th)/2 = 13, so 3rd + 4th = 26
Step 4: The three blanks must be counting numbers that allow this condition
Step 5: Multiple combinations possible - count all valid arrangements