Question

Five integers are picked from 0 to 20, with possible repetitions, such that their mean is 12, median is 18, and they have a single mode of 20.

Ignoring permutations, the number of ways to pick these five integers is _____

• A. 0
• B. 1
• C. 2
• D. 3

Hint:

The median fixes the middle number at 18, so 20 can only sit in the fourth and fifth positions, meaning it can appear at most twice.

Answer 1

The Correct Option is B

Step 1: Set up the sorted list and use the mean.
Arrange the five chosen integers in non-decreasing order as \(a \le b \le c \le d \le e\), with every value between 0 and 20. The mean is 12, so the sum of all five numbers is \(5 \times 12 = 60\).

Step 2: Use the median to fix the middle value.
The median of five sorted numbers is the middle (third) value, so \(c = 18\). Since the list is sorted, this also forces \(a \le b \le 18\) and \(18 \le d \le e \le 20\).

Step 3: Use the mode condition to pin down d and e.
We need 20 to be the single (unique) mode, meaning 20 appears strictly more often than every other value in the list. Since \(a, b \le c = 18\), the value 20 can only appear among \(d\) and \(e\), so 20 can appear at most twice. If 20 appeared only once, no value in the list could stand out as a repeated, unique mode (if every value were distinct there would be no single mode at all). So for 20 to be a genuine, unique mode, it must appear exactly twice, which forces \(d = e = 20\).

Step 4: Solve for a and b.
With \(c = 18\), \(d = e = 20\), the sum condition gives \(a + b + 18 + 20 + 20 = 60\), so \(a + b = 2\). Since \(a \le b \le 18\) and both are non-negative integers, the only possibilities are \((a,b) = (0,2)\) or \((a,b) = (1,1)\).

Step 5: Apply the single-mode condition to rule out ties.
If \((a,b) = (1,1)\), the value 1 would also appear exactly twice, tying with 20's frequency of two, so 20 would no longer be the unique mode. This case is invalid.
If \((a,b) = (0,2)\), the values 0, 2 and 18 each appear once, and only 20 appears twice, so 20 is genuinely the single mode. This case is valid, giving the multiset \(\{0, 2, 18, 20, 20\}\).

Step 6: Count the valid selections.
Only one multiset of five integers satisfies every condition (mean 12, median 18, single mode 20). Since the question asks to ignore permutations, that is, count unordered selections and not orderings, the answer is exactly one way.

Final Answer:
There is exactly 1 way to pick the five integers: \(\{0, 2, 18, 20, 20\}\). \[ \boxed{1} \]