A quadruple (a, b, c, d) of distinct integers is said to be balanced if a + c = b + d .
Let S be any set of quadruples (a, b, c, d) where 1 ≤ a < b < d < c ≤ 20 and where
the cardinality of S is 4411. Find the least number of balanced quadruples in S 

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.

A quadruple (a, b, c, d) of distinct integers is said to be balanced if a + c = b + d. Let S be any set of quadruples (a, b, c, d) where 1 ≤ a < b < d < c ≤ 20 and where the cardinality of S is 4411. Find the least number of balanced quadruples in S.

Solve the combinatorial problem of finding the minimum number of balanced quadruples (a, b, c, d) where 1 ≤ a < b < d < c ≤ 20. This solution explains key combinatorial concepts, including the binomial theorem and balanced quadruple conditions, with step-by-step calculations for determining the least number of balanced quadruples.

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