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:

Fix the median at 18, then check where a repeated 20 can go so the sum still gives mean 12, and test the single-mode condition on each case.

Answer 1

The Correct Option is B

Step 1: Set up the sorted list and use the mean.
Since we only care about the multiset of five integers (order does not matter), arrange them as \(a_1 \le a_2 \le a_3 \le a_4 \le a_5\), each an integer from \(0\) to \(20\).
The mean is \(12\), so the sum of all five integers is:
\[ a_1+a_2+a_3+a_4+a_5 = 5 \times 12 = 60 \]

Step 2: Use the median.
For five numbers sorted in order, the median is the middle (third) value, so:
\[ a_3 = 18 \]
This forces \(a_1 \le a_2 \le 18 \le a_4 \le a_5 \le 20\), since everything from the third position onward must be at least \(18\) and everything up to it at most \(18\).

Step 3: Use the mode condition to pin down \(a_4\) and \(a_5\).
A "single mode of 20" means \(20\) must appear more often than every other value, so \(20\) must occur at least twice among the five numbers.
Since \(a_1, a_2, a_3 \le 18\), which is less than \(20\), only \(a_4\) and \(a_5\) can possibly equal \(20\). To get two copies of \(20\), we need:
\[ a_4 = a_5 = 20 \]

Step 4: Solve for the remaining two numbers.
Substitute \(a_3=18\), \(a_4=20\), \(a_5=20\) into the sum equation:
\[ a_1 + a_2 + 18 + 20 + 20 = 60 \]
\[ a_1 + a_2 = 2 \]
with \(0 \le a_1 \le a_2 \le 18\). The possible integer pairs are \((0,2)\) and \((1,1)\).

Step 5: Apply the "single mode" condition to eliminate one case.
Case \((1,1)\): the five numbers become \(1,1,18,20,20\). Here \(1\) occurs twice and \(20\) occurs twice, so there are two modes, not a single mode. This violates the requirement that \(20\) alone is the mode, so this case is rejected.
Case \((0,2)\): the five numbers become \(0,2,18,20,20\). Here every value except \(20\) occurs exactly once, and \(20\) occurs twice, more than any other value, so \(20\) is the single mode. Check: mean \(= (0+2+18+20+20)/5 = 60/5 = 12\), median \(=18\) (the third value when sorted), single mode \(=20\). All conditions hold.

Step 6: Final Answer.
Only one multiset, \(\{0, 2, 18, 20, 20\}\), satisfies all three conditions. Ignoring permutations, there is exactly one way to pick these five integers. \[ \boxed{1} \]