Determining Integer Combinations from Mean, Median, and Mode

We need to find the number of ways to select five integers from the range 0 to 20, with repetitions allowed. The chosen set must satisfy: mean = 12, median = 18, and a single mode = 20.

Applying the Median Condition

Let the five integers, sorted in non-decreasing order, be $x_1, x_2, x_3, x_4, x_5$. The median is the middle value, $x_3$. Given the median is 18:

$ x_3 = 18 $

Applying the Single Mode Condition

The mode is the most frequent value. A single mode of 20 means 20 appears most often, and no other number appears as frequently.

Since the integers are sorted ($x_1 \le x_2 \le x_3 \le x_4 \le x_5$) and $x_3 = 18$, the mode 20 must occur in the later positions. Therefore, $x_4 = 20$ and $x_5 = 20$.

Because 20 appears twice and must be the *single* mode, no other integer can appear twice. This means $x_1$ and $x_2$ must be distinct from each other and also distinct from 18. The constraints become $0 \le x_1 \le x_2 \le 18$, with $x_1 \neq x_2$, $x_1 \neq 18$, and $x_2 \neq 18$.

Applying the Mean Condition

The mean is given as 12. The sum of the five integers is:

$ \frac{x_1 + x_2 + x_3 + x_4 + x_5}{5} = 12 $

$ x_1 + x_2 + x_3 + x_4 + x_5 = 5 \times 12 = 60 $

Substitute the known values ($x_3=18, x_4=20, x_5=20$):

$ x_1 + x_2 + 18 + 20 + 20 = 60 $

$ x_1 + x_2 + 58 = 60 $

$ x_1 + x_2 = 2 $

Evaluating Possible Integer Combinations

We need pairs $(x_1, x_2)$ satisfying:

• $x_1 + x_2 = 2$
• $0 \le x_1 \le x_2 \le 18$
• $x_1 \neq x_2$ (for a single mode of 20)

Possible pairs $(x_1, x_2)$ summing to 2 ($x_1 \le x_2$, $x_1, x_2 \ge 0$):

1. Pair 1: $(x_1, x_2) = (0, 2)$

   Conditions Check: $0+2=2$. $0 \le 2 \le 18$. $0 \neq 2$. $0 \neq 18$, $2 \neq 18$. All conditions are met.

   Resulting Set: {0, 2, 18, 20, 20}. Frequencies: 0 (1), 2 (1), 18 (1), 20 (2).

   Single Mode: 20. This combination is valid.

2. Pair 2: $(x_1, x_2) = (1, 1)$

   Conditions Check: $1+1=2$. $1 \le 1 \le 18$. However, $x_1 = x_2$.

   Resulting Set: {1, 1, 18, 20, 20}. Frequencies: 1 (2), 18 (1), 20 (2).

   This set has two modes (1 and 20), violating the *single* mode requirement. This combination is invalid.

Conclusion on Number of Ways

Only one combination, {0, 2, 18, 20, 20}, satisfies all the given conditions: mean=12, median=18, single mode=20, and integers within the 0-20 range.

Thus, the number of ways to pick these five integers is 1.