# How many $n\times n$ binary matrices are there, such that at least one row is filled by only $0'$s? [closed]

How many $$n\times n$$ binary matrices (values are only $$0'$$s and $$1'$$s) are there, such that at least one row is filled by only $$0'$$s?

How to approach this problem?

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The number of binary matrices with at-least one "zero" row $$=$$ Total no. of binary matrices $$-$$ No. of binary matrices with no "zero" row
Total no. of binary matrices $$=2^{n^2}\because$$ the matrix contains $$n^2$$ entries each of which can be $$0,1$$.
For the second part, observe each row of the binary matrix can be set up in $$2^n$$ ways by virtue of having $$n$$ entries that may be $$0,1$$. Each row is a non-"zero" row, so out of the $$2^n$$ possibilities for a given row, all but one (that is the the "zero" row) are admissible. So, each row can be set up in $$2^n-1$$ ways. Since there are $$n$$ rows, no. of binary matrices with no "zero" row $$=(2^n-1)^n$$.
This answer is equal to $$2^{n^2}-(2^n-1)^n$$.