# Getting The Lights Turned On

Probability Level 5

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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