12 ingredients must be available.
It is given that the restaurant wants to offer over 4000 different omelets.
[tex]2^{x} =4000[/tex], where x is the minimum number of ingredients.
[tex]x\ln \left(2\right)=\ln \left(4000\right)[/tex]
[tex]x=\frac{\ln \left(4000\right)}{\ln \left(2\right)}[/tex]
[tex]=11.96578\dots[/tex]
So, 12 ingredients must be available.
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