Stars and Bars (Medium)

AMC 10/12 mid (≈Q9–16) — multi-step.

Stars and Bars: core idea

Positive via

y_i=x_i-1

How many **nonnegative** solutions does $x_1+x_2+x_3+x_4=10$ have?

Hint: $\binom{10+4-1}{4-1}$ . Answer: 286.

1. How many **positive** solutions does $x_1+x_2+x_3=11$ have?
2. How many nonnegative solutions to $x+y+z=15$ ?
3. How many **positive** solutions to $x_1+x_2+x_3+x_4=10$ ?
4. Nonnegative solutions to $x_1+x_2+x_3=12$ with $x_1\ge 3$ ?
5. How many **positive** solutions to $x_1+x_2=17$ ?
6. Distribute $8$ identical balls into $5$ distinct bins (empty bins allowed). How many ways?
7. How many **positive** solutions to $x_1+x_2+x_3=14$ ?
8. How many nonnegative solutions to $x_1+x_2=20$ ?
9. Distribute $12$ identical tokens into $4$ distinct boxes (some may be empty). How many ways?