Combination problem distributing 
In how many ways 8 candies can be distributed be among 3 children?

I have done by using combination method $8C4$ but the answer is given 45. 
someone has done in this way: 





*

*008 in 3 ways 

*710 in 6 ways

*611 in 3 ways

*620 in 6 ways

*530  in 6 ways

*521 in 6 ways

*440 in 3ways

*431 in 6 ways

*422 in 3   ways

*332 in 3 ways
in total 45 ways.
Why my formulation is not right.  And how can I do this math within the formula I have writen? I mean within short time done.

If you're planning on using the stars and bars method, it is given by $$\binom{8+3-1}{8}=45.$$
We have eight candies (or stars) and two bars that separate the candies.  E.g. $$\star\star|\star\star\star\star\star|\star$$ corresponds to the children receiving $2$, $5$, and $1$ candies, respectively.  And there are $\binom{8+3-1}{8}$ inequivalent ways to sequence the stars and bars.
