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(I have seen a similar question for n×n matrix. However the answer to it just points to an a link (http://oeis.org/A001499) without explaining how to get the result through combinatorics)

How do i find the number of a 4×4 matrix with each row and each column having two 1s and two 0s?

My approach:

I noted that 0011 0011 1100 1100

Follows the rule.

So i tried the fact that swapping any two rows and/or any two columns also satisfies the result.

But in this approach i am getting a lot of repetitions i cant handle.

Sage of Seven Paths
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    Do you want to find all such $4\times4$ matrices, count them, just find a specific example (which you've done), etc. ? What is the actual question you'd like answered because I'm not sure. –  Sep 15 '22 at 11:04
  • HINT: This question can be solved using P.I.E such that firstly place the ones into each rows such that $C(4,2)^4=6^4 =1296$ ways. There are $1296$ possible matrix which have $2$ ones (and so zeros) in each rows. However , it contains some unwanted configurations such that the columns contains more than $2$ ones or less than $2$ ones. Here , the P.I.E gets in the deed (and a little cumbersome for this question) – Not a Salmon Fish Sep 15 '22 at 11:46
  • Why don't you say "yes" to the question of Logan M ? – Jean Marie Sep 15 '22 at 13:57
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    @Salmon Fish What is P.I.E. (Principle of Independent Events ?) ? – Jean Marie Sep 15 '22 at 13:59
  • This question : math.stackexchange.com/q/35019/305862 is about the number of such matrices ; in particular, the accepted answer is connected to the site you mention oeis.org/A001499 – Jean Marie Sep 15 '22 at 14:18
  • @LoganM yes i made a pathetic error of language – Sage of Seven Paths Sep 15 '22 at 15:34
  • @JeanMarie yes indeed ot is the question i was talking about. I am trying to find the answer with a particular value of n – Sage of Seven Paths Sep 15 '22 at 15:38

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