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Combinatorics (tổ hợp)
Factorial: $n!$
Combinations (regardless of order):
Taken k objects at a time from n different objects (Tổ Hợp chập k của n):
\[\mathbf{C_{n}^{k}} = \binom{k}{n} = \frac{n!}{k!(n-k)!}\]Properties:
$C_{n}^{k} = C_{n}^{n-k}$
$C_{n-1}^{k-1} + C_{n-1}^{k} = C_{n}^{k}$
$C_{n}^{0} + C_{n}^{1} + C_{n}^{2} + … + C_{n}^{n} = 2^n$
Permutations (distinction in order):
- Taken n different objects (Hoán Vị): $\mathbf{P_{n}} = A_{n}^{n} = n!$
- Taken k objects at a time from n different objects (Chỉnh Hợp chập k của n):
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Taken n objects not all different: $\frac{n!}{n_{1}!n_{2}!n_{3}!…n_{k}!}$
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Taken n different objects arranged in a circle: $(n-1)!$
The Inclusion-Exclusion Principle: For any two sets $A$ and $B$,
\[P(A\cup B) = P(A) + P(B) - P(A\cap B)\]The Complement Principle: If set $A$ is a subset of a universal set $U$, then
\[P(A) = P(U) - P(A^C)\]where the complement of the event $A$, consisting of all elements in the sample space $S$ that are not elements of the set $A$, is denoted by $A^C$
