269: Probability Sums

5 distinct numbers are chosen at random from {1,2,3,4,5,6,7,8,9}.

p(k) = probability their sum = k.

What are some of the ways you can find this in general?
What sum is/are the least likely?
Which sum is/are most likely?

p(15)=?

p(35)=?

source:

5 distinct numbers are chosen at random from {1,2,3,4,5,6,7,8,9}. p(k) = probability their sum = k. p(15)=?,..., p(35)=?
— Republic of Math (@republicofmath)

and

Five distinct nmbrs chosen at random from {1,2,3,...,9} and added together. What is the most likely sum?
— James Tanton (@jamestanton)