Problem
COM-B2-M01-P003 A Marked Line
#3
★★☆☆☆ Level 2 of 5
In a \(12\times12\) table, \(70\) cells are marked. Prove that some row or column contains at least \(6\) marked cells.
Each marked cell lies in one row and one column.
Count pairs \((cell, line)\), where the line is a row or column containing the marked cell. Each marked cell gives \(2\) pairs, so there are \(140\). There are \(24\) lines. The average is \(\frac{140}{24}>5\), so some line contains at least \(6\) marked cells.
It is useful to stress why the conclusion is “row or column”, not just row.