Combinations (Medium)

AMC 10/12 mid (≈Q9–16) — multi-step.

Combinations: core idea

Identities.

How many ways can a $4$ -person committee be chosen from $10$ people?

Hint: Split restricted selections into independent choices. Answer: 210.

1. How many diagonals does a decagon have?
2. How many ways can $3$ red balls be chosen from $7$ red balls?
3. How many $5$ -card hands can be chosen from $10$ cards?
4. How many ways can $3$ pizza toppings be chosen from $8$ ?
5. A class has $6$ boys and $5$ girls. How many $3$ -person groups have exactly two girls?
6. How many ways can $4$ vertices of a hexagon be selected?
7. How many ways can $2$ prizes be given to $6$ people if the prizes are identical?
8. How many ways can $5$ questions be selected from $12$ ?
9. How many ways can a team of $4$ be selected from $9$ people if one person must be included?