12 Smarties Question revisited?

Question

Suppose you have 12 identical Smarties and 3 friends. You want to divide them amongst 3 friends, so that each friend gets at least 3 smarties. How many ways can this be done?

But there may be remainder, and the possibilities can be flipped around because each person is distinct.

Answer 1

10 ways.

This problem is equivalent to finding the number of solutions in integers to:
x1 + x2 + x3 = 12
where xk > 2 . (k = 1, 2, 3)
This is C(12 - 2 - 2 - 2 - 1, 2) = C(5,2) = 10.
where C(a,b) = a! / [(a-b)!b!]

The actual values are:
3,3,6
3,4,5
3,5,4
3,6,3
4,3,5
4,4,4
4,5,3
5,3,4
5,4,3
6,3,3

If there can be a remainder (i.e., some left over after the 3 friends get at least 3 each), we need to add C(4,2)= 6,
C(3,2) = 3, and C(2,2)= 1. So the total is 20.

3,3,5
3,4,4
3,5,3
4,3,4
4,4,3
5,3,3 (these total 11)

3,3,4
3,4,3
4,3,3 (these total 10)

3,3,3 (this totals 9)

In general, the number of solutions in integers to the equation:
x1 + x2 + x3 + ... + xk = m
where x1>c1, x2>c2, x3>c3, ... xk>ck for some integers c1 through ck, is:

C(m - c1 - c2 - c3 - ... - ck - 1, k-1)

In the initial problem above, m = 12, k = 3, and ck=2 for all k.
For the subsequent problem allowing a remainder, we add the number of solutions when m=11, m=10 and m=9.

You may find that other answers simply enumerate the possibilities. While this method may be sufficient for this particular problem, it becomes impractical if there were say, 100 smarties and 10 friends
(205,811,513,765 solutions [ =C(79,9) ] with no remainder). Use of the equation is preferable.

Answer 2

This problem is made vastly easier since all the smarties are identical. If the smarties were unique, it would be quite complicated.

The number of ways to arrange the NUMBER of smarties that each friend gets can range anywhere from 3-3-6 (least equitable) to 4-4-4 (most equitable). The possible ways are
3-3-6 (in any order)
3-4-5 (in any order)
4-4-4 (only one order)

The number of unique ways to arrange 3, 3, and 6 is three:
3-3-6
3-6-3
6-3-3

The number of unique ways to arrange 3, 4, and 5 is six:
3-4-5
3-5-4
4-3-5
4-5-3
5-3-4
5-4-3

The number of ways to arrange 4, 4, and 4 is one:
4-4-4

Each of these is a possible way of distributing the identical smarties. There are thus
3 + 6 + 1 = 10 unique ways to distribute the smarties.

Answer 3

4 - 4 - 4
3 - 4 - 5
3 - 3 - 6

Looks like three combinations.