Colorings avoiding patterns

#IE#coloring#grid

Statement

Count $3\times3$ $0$ - $1$ matrices with no all- $1$ row.

Hint

IE on "row $i$ is all ones".

Show solutionHide solution

Sets. $A_i$ = row $i$ all ones. $|A_i|=2^6=64$ .

Pairs. $|A_i\cap A_j|=2^3=8$ .

Triple. $|A_1\cap A_2\cap A_3|=1$ .

IE. $512-3\cdot64+3\cdot8-1=512-192+24-1=343$ .

Conclusion. $343$ matrices have no all- $1$ row.