This is a classic bars and bars problem. Refer this link to understand this concept
https://www.youtube.com/watch?v=UTCScjoPymA
In context to this problem, this is how it is solved using stars and bars.
There are 10 identical sweets.
_ _ _ _ _ _ _ _ _ _
(In analogy to stars and bars, these are the stars)
Now, these 10 sweets are distributed among 3 children say A,B and C
Let's put 2 bars in between those places (Thereby dividing the 10 sweets into 3 sections. Each section determines how many sweets a child gets). Such that, to the left of bar 1 are sweets of A. Between the bars is sweets of B. Sweets to the right on bar 2 are C's.
Consider this one possible arrangement
_ _ _ | _ _ _ _ | _ _ _
Thus A gets 3 here, B gets 4 and C gets 3.
Now, these 2 bars can be anywhere (for 2nd case when you don't have any restriction).
So, give them slots too.
Now, we have 10+2 = 12 slots (In general n+k-1)
We have to choose 2 positions for bars (In general k-1)
Hence we have the formula
(They are equal due to combination property)
Similarly, by restricting the positions of bars (for case 1) you can find the answer.