Count the number of distinct matrices \( A \), where two matrices are considered identical if one can be obtained from the other by rearranging rows and columns, that have the following properties:

1. \( A \) is a \( 7 \times 7 \) matrix and every entry of \( A \) is \( 0 \) or \( 1 \).
2. Each row of \( A\) contains exactly 3 non-zero entries.
3. For any two distinct rows \( i\) and \( j\) of \( A\), there exists exactly one column \( k \) such that \( A_{ik} \neq 0 \) and \( A_{jk} \neq 0 \).

The best solution was submitted by 권오관 (연세대학교 수학과 22학번, +3). Congratulations!

Here is the best solution of problem 2024-12.

Other solutions were submitted by 김준홍 (KAIST 수리과학과 석박통합과정, +3), 김찬우 (연세대학교 수학과 22학번, +3), 노희윤 (KAIST 수리과학과 석박통합과정, +3), 양준혁 (KAIST 수리과학과 20학번, +3), 이명규 (KAIST 전산학부 20학번, +3), 정영훈 (KAIST 새내기과정학부 24학번, +3), 채지석 (KAIST 수리과학과 석박통합과정, +3), Eun Kyeol (+3).