Getting The Lights Turned On

A room contains 44 lightbulbs L1,L2,L3,L4L_1, L_2, L_3, L_4, and 44 switches S1,S2,S3,S4S_1, S_2, S_3, S_4. Flipping switch SiS_i will toggle the settings of exactly ii lightbulbs. All the lightbulbs begin in the off position. For each lightbulb, there is a combination of switches that we can flip which will result in only that lightbulb being on.

We create a 4×44 \times 4 table where ai,j,a_{i,j}, the entry in row ii and column j,j, is 11 if switch ii toggles bulb jj and 00 otherwise. How many different tables can we form?

Details and assumptions

As an explicit example, if switch SiS_i toggles the settings of lightbulbs 11 to ii, then the table would be

l1l2l3l4S11000S21100S31110S41111\begin{array}{c| cccc} & l_1 & l_2 & l_3 & l_4 \\ \hline S_1 & 1 & 0 & 0 & 0\\ S_2 & 1 & 1 & 0 & 0\\ S_3 & 1 & 1 & 1 & 0\\ S_4 & 1 & 1 & 1 & 1\\ \end{array}

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