The posted question is virtually the same as the following:
How many ways can 6 identical chocolates be distributed among 4 children?We can use the SEPARATOR METHOD.
6 identical chocolates are to be separated into -- at most -- 4 groupings.
Thus, we need 6 chocolates and 3 separators:
O|O|OO|OO
Every arrangement of the elements above represents one way to distribute the 6 chocolates among 4 children A, B, C and D:
O|O|OO|OO = A gets 1 chocolate, B gets 1 chocolate, C gets 2 chocolates. D gets 2 chocolates
This case is the equivalent of the following integer: 1122
||OO|OOOO = A gets 0 chocolates, B gets 0 chocolates, C gets 2 chocolates. D gets 4 chocolates
This case is the equivalent of the following integer: 0024 = 24
OOOOOOOOOO||| = A gets all 6 chocolates
This case is the equivalent of the following integer: 6000
To count the number of possible distributions, we need to calculate the number of ways to position the 3 separators.
Since there are 6 chocolates and 3 separators -- yielding a total of 9 positions -- the 3 separators must occupy 3 of the 9 positions.
From 9 positions, the number of ways to choose 3 to assign to the 3 separators =
9C
3 \(= \frac{9*8*7}{3*2*1} = 84 \)