Inclusion-Exclusion (Medium)

AMC 10/12 mid (≈Q9–16) — multi-step.

Inclusion-Exclusion: core idea

Three sets.

How many integers from $1$ to $1000$ are **not** divisible by $2$ , $3$ , or $5$ ?

Hint: Count divisible by $2,3,5$ with PIE, subtract from $1000$ . Answer: 266.

1. How many integers from $1$ to $100$ are divisible by $2$ **or** $3$ **or** $5$ ?
2. How many integers from $1$ to $40$ are divisible by $2$ **or** $3$ **or** $5$ ?
3. How many surjective functions are there from a $5$ -element set to a $3$ -element set?
4. How many length- $6$ binary strings contain **no** two consecutive $1$ s?
5. How many integers $1\le n\le100$ satisfy $\gcd(n,100)>1$ ?
6. How many integers from $1$ to $210$ are divisible by $2$ **or** $3$ **or** $5$ **or** $7$ ?
7. How many functions from a $4$ -element set to a $2$ -element set are **surjective**?
8. How many integers from $1$ to $360$ are divisible by $8$ **or** $9$ **or** $10$ ?
9. How many integers from $1$ to $30$ are divisible by **neither** $4$ **nor** $6$ ?