Combinations (Hard)

Late AMC 10/12 through early AIME — dense computation.

Combinations: core idea

Hockey-stick.

Choose a four-person committee from six women and five men with at least two women. How many?

Hint: Use cases, complements, or stars and bars as appropriate. Answer: 265.

1. How many diagonals has a $15$ -gon?
2. From $12$ students select president, vice president, and a disjoint three-person committee. How many?
3. How many three-element subsets of $\{1,\ldots,10\}$ contain no consecutive integers?
4. Distribute five identical candies to three children, each receiving one. How many ways?
5. How many right/up paths from $(0,0)$ to $(5,4)$ pass through $(2,2)$ ?
6. Choose five from ten people if two named people cannot both be chosen. How many?
7. How many nonnegative solutions has $x+y+z=12$ with $x\ge2$ ?
8. Ten distinct toppings: how many pizzas use at least one but not all toppings?
9. Put three identical red and four identical blue balls into two labeled boxes. How many ways?