Your requirement is not totally clear.
A. Do you just want to find any string that contains at least 1 each of a, n, u?
B. Do they have to be in the order of something, then 'a', then later an 'n', then later a 'u'?
Of your 4 examples, it appears that 1 and 4 should fail, and 2 and 3 should pass.
Right?
For the case A above:
(?:a.*n.*u)|(?:a.*u.*n)|(?:n.*a.*u)|(?:n.*u.*a)|(?:u.*n.*a)|(?:u.*a.*n)
For case B:
(?:a.*n.*u)
The / delimiters and i at the end for case insensitive depend on the context where you are using this. For example, these would be appropriate for a Perl script, but not for a .NET Regex.