To solve this problem, we need to find the minimum number of balanced quadruples in a set $S$ of quadruples $(a, b, c, d)$ where $1 \leq a < b < d < c \leq 20$ and the cardinality of $S$ is 4411.

Step 1: Count the total number of quadruples

Given the conditions $1 \leq a < b < d < c \leq 20$, we need to count the number of valid quadruples.

We can choose four distinct numbers from the set $\{1, 2, \dots, 20\}$ and assign them to $a, b, c, d$ in such a way that $a < b < d < c$. The number of ways to choose 4 distinct numbers from 20 is given by:

$\binom{20}{4} = \frac{20 \times 19 \times 18 \times 17}{4 \times 3 \times 2 \times 1} = 4845$

For each choice of four numbers, there is exactly one way to assign them to $a, b, d, c$ such that $a < b < d < c$. Therefore, the total number of valid quadruples is 4845.

Step 2: Establish balanced quadruples

A quadruple $(a, b, c, d)$ is balanced if $a + c = b + d$. To find the least number of balanced quadruples, we need to analyze the structure of the problem.

Step 3: Maximize non-balanced quadruples

The problem asks for the minimum number of balanced quadruples. Since $|S| = 4411$, we need to minimize the number of balanced quadruples in this set.

The total number of non-balanced quadruples can be at most $4845 - 4411 = 434$. So the minimum number of balanced quadruples is:

$4845 - 434 = 4411$

Therefore, to maximize non-balanced quadruples and minimize balanced ones, there must be at least:

$4411 - 434 = 3977$

Thus, the least number of balanced quadruples in $S$ is 3977.

Final Answer

The minimum number of balanced quadruples in the set $S$ is:

$\boxed{3977}$

Would you like a deeper explanation on any step, or have more questions?

Here are 5 relative questions you might find interesting:

1. How would the answer change if the quadruples were chosen from $\{1, 2, \dots, 30\}$?
2. How do we compute combinations when the constraints on elements differ, like $a \leq b \leq d \leq c$?
3. What properties do balanced quadruples share, and how can these be used in combinatorial problems?
4. How do we calculate the probability of randomly selecting a balanced quadruple from the set $S$?
5. Can this method be generalized to other types of quadruples, such as those involving different arithmetic conditions?

Tip: Understanding how to count distinct combinations in problems like these can be greatly enhanced by practicing with smaller sets first, where you can manually verify all possible cases.