How to Solve a Stars and Bars Discrete Math Problem

Let's look at stars and bars first, with $15$ Tootsie rolls.
One way of distributing them to her $3$ brothers could be

$Largeboxed.boxed.boxed.+boxed.boxed.boxed.boxed.boxed.+boxed.boxed.boxed.boxed.boxed.boxed.boxed.= 15$

The Tootsie rolls (n) are "stars" and the $+'s$ are "bars", and it should be clear that we can get all the ways to distribute by just deciding where to put the $+'s$, which can be done in $binom{17}2$ ways.

Note for reference that the number of $+'s$ (bars) will always be $1$ less than the number of children (k), thus the generalised formula $binom{n+k-1}{k-1}$

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For part $1$, we distribute the two types separately (independently), and multiply, viz $binom{17}{2}binom{12}{2}$

For part $2$, we pre-distribute $3$ Tootsie rolls and $3$ Twizzlers, (one each) and distribute the balance using stars and bars, as before.

1) If there were only 15 items on type 1 without restrictions, all you need to do ot distribute among 4 people is to see that it is in fact just $binom{17}{3}$ because there are 3 bars 2) I this case just subtract 3 from each type and repeat the calculations in 1)