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Problems

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Counting

Combinatorics / inclusion-exclusion

Stars and bars, nCr identities, |A u B| = |A| + |B| - |A n B|.

Quiz (10) Spot the pattern
Notes
  • Stars and bars: distributing n identical items into k boxes gives C(n + k - 1, k - 1) ways.
  • Permutations of a multiset: n! / (c1! c2! ...).
  • Inclusion-exclusion: |A ∪ B ∪ C| = Σ|A| - Σ|A∩B| + |A∩B∩C|. Used to count "at least one", or derangements (!n ≈ n!/e).
  • Pascal: C(n, k) = C(n-1, k-1) + C(n-1, k). Use it for small n or when there's no prime modulus.

4 problems

Advanced & competitive

Counting Combinatorics, inclusion-exclusion, Catalan numbers.

Combinatorics / inclusion-exclusion

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