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Advanced Counting Problems

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Now · Circular Permutations

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.

Example: Arrange 5 people in a line such that A and B are not adjacent. Total arrangements:

5! = 120

Arrangements where A and B are adjacent: treat AB as one unit →

4! \times 2 = 48

Arrangements where A and B are NOT adjacent:

120 - 48 = 72

$\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}$ (Pascal)
$\sum_{k=0}^n \binom{n}{k} = 2^n$
$\binom{n}{k} = \binom{n}{n-k}$
$\binom{n}{0} + \binom{n}{2} + \binom{n}{4} + \cdots = 2^{n-1}$ Join Discord PreviousAdvanced CountingNextPigeonhole & Inclusion-Exclusion