Five integers are picked from 0 to 20 with possible representation such that their mean is 12, median is 18 and they have single mode of 20. Ignoring the permutation, the number of ways to pick these five integers is ____ .

The correct option is 1.

Let the five integers chosen from the range 0 to 20 be represented in non-decreasing order as:
$x_1\le x_2\le x_3\le x_4\le x_5$
where each .

We are given the following conditions:
1. Median: The median of five numbers in sorted order is the middle (third) number. Thus,
$x_3=18$.
2. Mode: The single (unique) mode is 20. For 20 to be a mode, it must appear more than once. Since 20 is the maximum possible value in the range [0, 20], and $x_3=18$, the value 20 can only occupy the positions after the median, which are $x_4$ and $x_5$. Since the mode must be unique and appear at least twice, we must have:
$x_4=x_5=20$.
(If only one of them were 20, it wouldn't be a mode unless all five numbers were distinct, but then there would be no mode or all would be modes. Since it has a single mode of 20, 20 must appear at least twice. Thus, $x_4=20$ and $x_5=20$).
3. Mean: The mean of the five integers is 12. Therefore, the sum of the five integers is:
$x_1+x_2+x_3+x_4+x_5=12\times 5=60$.

Substituting the known values $x_3=18$, $x_4=20$, and $x_5=20$ into the sum equation:
$x_1+x_2+18+20+20=60$
$x_1+x_2+58=60$
$x_1+x_2=2$.

We must also respect the sorted order and boundary constraints:
$0\le x_1\le x_2\le x_3=18$.

Let's find the integer pairs $(x_1,x_2)$ that satisfy $x_1+x_2=2$ and $0\le x_1\le x_2$:
- Case 1: $x_1=0$, which gives $x_2=2$. The set of integers is $\{0,2,18,20,20\}$. Here, 20 is the unique mode (frequency 2, others frequency 1).
- Case 2: $x_1=1$, which gives $x_2=1$. The set of integers is $\{1,1,18,20,20\}$. However, in this case, both 1 and 20 appear twice, which means there would be two modes (1 and 20). This violates the requirement of having a single mode of 20.

Thus, only Case 1 is valid, giving the unique set of integers: $\{0,2,18,20,20\}$.
Consequently, ignoring permutation, there is only 1 way to pick these five integers.