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Factorial, Permutations, Combinations (distinct objects; no repeats)

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1) n! (factorial) Meaning: tells you how many permutations (complete orderings) you can generate with n objects at your disposal. Definition: n! = n × (n−1) × (n−2) × … × 2 × 1, with 0! = 1. Example: 5! = 5×4×3×2×1 = 120. 2) n!/(n−r)! (permutations of r choices from n; order matters) Meaning: tells you how many ordered outcomes you can generate when you make r choices out of a collection of n objects, without reuse. How to see it: 1st choice: n options 2nd choice: (n−1) options 3rd choice: (n−2) options … rth choice: (n−r+1) options Multiply: n × (n−1) × … × (n−r+1) = n!/(n−r)!. Example (n=5, r=2): 5P2 = 5!/(5−2)! = 5!/3! = (5×4×3×2×1)/(3×2×1) = 5×4 = 20. 3) (n!/(n−r)!)/r! = n!/((n−r)! r!) (combinations; order neglected) Meaning: tells you how many selections you can make when choosing r objects from n, where order does not matter. ...