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

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

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