A father has nine identical coins to give to his three children. How many total allocations are possible?

Question

There's three parts to this question:

How many total allocations are possible? (This one I understand -- it's ${11 \choose 9}$ because it's unordered with replacement.)

How many allocations are possible if each child must receive at least one coin?

How many are possible if each child must receive at least two coins?

I have the answers but I have no idea how to explain them without counting out all the options that don't work and subtracting them from the total.

Answer 1

Each one of the other two questions can be easily reduced to the first one:

• If each child must receive at least $1$ coin, then solve the first question with $9-3\cdot1$ coins
• If each child must receive at least $2$ coins, then solve the first question with $9-3\cdot2$ coins

Answer 2

If each child must receive one coin, then give each child one coin and figure out how many ways there are to distribute the remaining $6$ coins. Similarly if each child must have at least two coins, begin by giving them all exactly two coins.

Note: If the father must give away all his coins, then this is a typical stars and bars problem. For the total number of allocations, it would be equivalent to placing $2$ bars between $12$ objects which gives $11\choose2$ possibilities.