Paragraph
  1. \({8 \choose 5}\) ways, after giving one present to each kid, you are left with 5 presents (stars) which need to be divide among the 4 kids (giving 3 bars).

  2. \({12 \choose 9}\) ways. You have 9 stars and 3 bars.

  3. \(4^9\text{.}\) You have 4 choices for whom to give each present. This is like making a function from the set of presents to the set of kids.

  4. \(4^9 - \left[{4 \choose 1}3^9 - {4\choose 2}2^9 + {4 \choose 3}1^9 \right]\) ways. Now the function from the set of presents to the set of kids must be surjective.

in-context