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n factorial, denoted as $n!$, is the product of all positive integers from 1 to n. It is used in permutations, combinations, and other areas involving sequences.
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Permutation: An arrangement of objects in a specific order. Calculated using $${}_nP_k = \frac{n!}{(n-k)!}$$.
Combination: A selection of items without regard to order. Calculated using $${}_nC_k = \frac{n!}{k!(n-k)!}$$.
Gamma Function: An extension of the factorial function that works for non-integer values. Denoted as $\Gamma(n)$, where $(n-1)! = \Gamma(n)$.