Exercise 886Combinatorics › Counting principlesL1Added 3 Nov 2026

Counting configurations

#inclusion-exclusion#configurations#chessboard

Statement

How many $3\times3$ $0$ - $1$ matrices have at least one row of all $1$ s?

Hint

Use inclusion-exclusion on "row $i$ is all $1$ s".

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Total. $2^9=512$ .

Count via IE. Let $A_i$ =row $i$ all ones. $|A_i|=2^6$ (other $6$ entries free). $|A_i\cap A_j|=2^3$ , $|A_1\cap A_2\cap A_3|=1$ .

Union. $3\cdot64-3\cdot8+1=192-24+1=169$ .

Conclusion. $169$ matrices have at least one all- $1$ row.