Getting The Lights Turned On

A room contains \(4\) lightbulbs \(L_1, L_2, L_3, L_4\), and \(4\) switches \(S_1, S_2, S_3, S_4\). Flipping switch \(S_i\) will toggle the settings of exactly \(i\) 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 \(4 \times 4\) table where \(a_{i,j},\) the entry in row \(i\) and column \(j,\) is \(1\) if switch \(i\) toggles bulb \(j\) and \(0\) otherwise. How many different tables can we form?

Details and assumptions

As an explicit example, if switch \(S_i\) toggles the settings of lightbulbs \(1\) to \(i\), then the table would be

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