Quiz 2

Advanced Counting Problems

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Python Week 1: the first filter for runtime behavior
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# Advanced Counting Problems ## Circular Permutations Number of ways to arrange $n$ distinct objects in a circle: $(n-1)!$ **Reasoning:** Rotations are considered identical, so fix one object and arrange the remaining $n-1$ objects relative to it. ## Permutations with Restrictions **Example:** Arrange 5 people in a...

Advanced Counting Problems

Circular Permutations

Number of ways to arrange nn distinct objects in a circle: (n−1)!(n-1)! Reasoning: Rotations are considered identical, so fix one object and arrange the remaining n−1n-1 objects relative to it.

Permutations with Restrictions

Example: Arrange 5 people in a line such that A and B are not adjacent. Total arrangements: 5!=1205! = 120 Arrangements where A and B are adjacent: treat AB as one unit → 4!×2=484! \times 2 = 48 Arrangements where A and B are NOT adjacent: 120−48=72120 - 48 = 72

Combinatorial Identities

  • (nk)=(n−1k−1)+(n−1k)\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k} (Pascal)
  • ∑k=0n(nk)=2n\sum_{k=0}^n \binom{n}{k} = 2^n
  • (nk)=(nn−k)\binom{n}{k} = \binom{n}{n-k}
  • (n0)+(n2)+(n4)+⋯=2n−1\binom{n}{0} + \binom{n}{2} + \binom{n}{4} + \cdots = 2^{n-1} Join Discord PreviousAdvanced CountingNextPigeonhole & Inclusion-Exclusion
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