Question: I am preparing for a competitive exam. Sorry if the question sounds naive.

The Below example was listed when I was studying **Fundamental Principle of Counting** which is

> 
> If you must make a number of separate decisions, then MULTIPLY the
>  number of ways to make each individual decision to find the number of
>  ways to make all the decisions
> 
> 
> 

The below question was not solved using the above technique,instead they listed all the possibilities.

My question is why the below problem cannot be solved using the above technique?

If yes, then how?

**Question**:

In how many ways 6 identical coins can be distributed among Alex,Bea and Chad?
**NOTE:** Some people may recieve zero coins?

Answer is **28**

Answer: 
I think the answer is $n+k-1\choose n $

where n= 6 , and k=3
since, we have no limit about how much coin we can gave to each person !
Here's a solution using the FPC:

Imagine Alex gets $a$ coins. The problem is then how many ways can we divide $6-a$ coins between Bea and Chad, which is a simple partition.

Imagine the $o$'s are coins:

$$o\;o\;o\;o\;o\;o$$

There are $7$ possible partition places, so for an example if $a=0$ and we have $oo|oooo$, then the distribution of coins is Alex:$0$, Bea:$2$ and Chad:$4$.

Next we observe that if Alex has $a$ coins, then there are $7-a$ possible partitions of the $6-a$ remaining coins. This allows us to group the number of coins Alex gets into $\{0\}, \{1,6\}, \{2,5\}, \{3,4\}$, with a total of $7$ possible partitions in each, hence the answer, by the FPC, is $4\times7=28$.