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Inclusion-Exclusion (Hard)

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

EasyMediumHardAIME / USAJMO / AMO

Inclusion-Exclusion: core idea

Complements / derangements.

Example problem: hint and answer

Integers 1≤n≤10001\le n\le1000 not divisible by 2,3,2,3, or 55?

Hint: PIE. Answer: 266.

Practice (9 problems — type your answers)

  • 1. Derangements !6!6.

  • 2. Surjections [6]→[4][6]\to[4].

  • 3. Positive integers ≤210\le210 divisible by 22 or 33 or 55 or 77?

  • 4. n≤100n\le100 with gcd⁡(n,100)>1\gcd(n,100)>1. Count?

  • 5. !5!5.

  • 6. Length-66 binary strings with no consecutive 11s.

  • 7. n∈[1,360]n\in[1,360] divisible by 88 or 99 or 1010.

  • 8. Functions [5]→[3][5]\to[3] that miss at least one value.

  • 9. Integers 1..301..30 divisible by neither 44 nor 66.

Inclusion-Exclusion Practice Problems — Hard