kawal27 wrote:
In how many ways can five identical balls be distributed in three different boxes?
A)15
B)18
C)20
D)21
E)24
The following is called the SEPARATOR method.
Five identical balls are to be separated into -- at most -- 3 groupings.
Thus, we need five balls and two separators:
OO|OO|O
Every arrangement of the elements above represents one way to distribute the 5 balls among three boxes A, B and C:
OO|OO|O = A gets 2 balls, B gets 2 balls, C gets 1 ball.
OO||OOO = A gets 2 balls, B gets 0 balls, C gets 3 balls.
OOOOO|| = A gets all 5 balls.
And so on.
To count all of the possible distributions, we simply need to count the number of ways to arrange the 7 elements above (the 5 identical balls and the 2 identical separators).
The number of ways to arrange 7 elements = 7!.
But when an arrangement includes identical elements, we must divide by the number of ways each set of identical arrangements can be arranged.
The reason:
When the identical elements swap positions, the arrangement doesn't change, reducing the total number of unique arrangements.
Here, we must divide by 5! to account for the 5 identical balls and by 2! to account for the 2 identical separators:
\(\frac{7!}{5!2!} = 21\)