(In reply to solution
Your result is ok , but the reasoning is redundant
6 - (5/6)^n - (4/6)^n - (3/6)^n - (2/6)^n - (1/6)^n
this equation states that the highest number is 6 unless the choise was made out of the first 5 numbers or only out of the first 4 numbers or... etc.. just all ones.
I got this result by plain reasoning without all the calculations and simplifications/
Hope you agree.