Stars and Bars (Easy)

AMC 10 Q1–8 depth — contest entry, not school drills.

Stars and Bars: core idea

Nonnegative:

\binom{n+k-1}{k-1}

How many **nonnegative** integer solutions does $x_1+x_2+x_3=8$ have?

Hint: $\binom{8+3-1}{3-1}=\binom{10}{2}$ . Answer: 45.

1. How many **positive** integer solutions does $x_1+x_2=10$ have?
2. How many ways can $7$ identical apples be split among $3$ children (each may get zero)?
3. How many **nonnegative** solutions to $x_1+x_2+x_3+x_4=6$ ?
4. How many **positive** solutions to $x_1+x_2+x_3=9$ ?
5. Distribute $5$ identical balls into $4$ distinct bins (empty bins allowed). How many ways?
6. How many **nonnegative** solutions to $x+y=12$ ?
7. How many **positive** solutions to $x_1+x_2+x_3+x_4=12$ ?
8. How many triples of nonnegative integers sum to $10$ ?
9. How many ordered pairs of **positive** integers have sum $15$ ?