A person throws a normal 6-sided dice a certian number of times, and their score is the highest value they throw. What is the probability of them getting a certain score?

I need to calculate the probability of getting all the scores 1 to 6, for up to about 5 throws. I can of course do this manually in a spreadsheet by entering an individual formula into every cell, but is there a more general formula I could use? I've tried looking at my notes but although I generally understand probabilities and permutations/combinations, I'm pretty stuck...

Any help would be really appreciated!