Customers choose candies with different flavours 
Each of the eight customers are choosing randomly and independently on each other one flavour of candy. In the bowl there many candies with five different flavours.
a) What is the probability that each flavour is chosen? And what would happen, if there were 100 customers?
b) What is the probability that all of the customers has agreed on the same flavour?

a) $P=1 - (\frac{1}{5})^{8} + (\frac{2}{5})^{8} + (\frac{3}{5})^{8} + (\frac{4}{5})^{8}$
My intuition is that one wasn't chosen , two wasn't chosen, three wasn't chosen ...
Is it correct?
For 100 customers it is the same just exponent is equal to 100.
b) It is just $P=(\frac{1}{5})^8$ right?

I think in the first case it should be $$P=1-\binom 51(\frac{4}{5})^8+\binom 52(\frac{3}{5})^8-\binom 53(\frac{2}{5})^8+\binom 54(\frac{1}{5})^8$$
Here I have utilised the Inclusion Exclusion Principle where the property is taken as a particular flavour is not chosen.
b) is incorrect.
Edit after @lulu comment to b)below the correct answer should be $5(\frac{1}{5})^8$.Read @lulu comment below for explanation.
