Quant Question No.1

Here is a question which was a bit ambiguous and several found difficult.
Try it out.

Q. Find the no of ways in which 14 identical and indistinguishable balls can be divided into 3 groups?

1) 16
2) 17
3) 18
4) 19
5) 20

My approach:
there are 14 balls which are to be divided into 3 groups.

It is not mentioned that the the groups are distinct or the balls are to be placed in 3 different sacks or something like that.

It implies that we are just supposed to separate the total of 14 balls into 3 different groups and the groups are indistinguishable as well(Mentioned this as i saw a lot of confusion on the earlier pages).

So now we separate the balls into 3 groups in the following manner:

1. (1,1,12)(1,2,11)(1,3,10)....(1,6,7) ....6 ways
2. (2,2,10)(2,3,9)(2,4,8,)(2,5,7)(2,6,6)...5 ways

• Note that i have not considered the case of (2,1,11) as it will be similar to the one considered in the previous iteration.

3. (3,3,8,)(3,4,7)(3,5,6)......................3 ways
4. (4,4,6)(4,5,5)...............................2 ways
5. (6,6,2)........................................It should not be considered as already considered in iteration 2.

Add up and you get a total of 16 different ways of representing the 14 balls into 3 groups.