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Number of different arrangements of n objects in a row
n!

Number of different arrangements when objects are repeated
TotalObjects! / (Number!Of!Each!Repeated!)

Picking groups, where order doesn't matter
(If 3 of 7 standby passengers are selected for a flight, how many diff. combos of standby passengers can be selected?)
Factorial of the total / (Factorial of group 1 * Factorial of group 2)
7! / (3!x4!)

Permutation where order matters, repetition is allowed
(# of permutations of a 3 digit lock?)
n^r
(3^3, or 3x3x3)

Order matters, no repetition allowed
(How many different orders of 3 balls selected from 16?)
n! / (nr)!  choose r things out of n
16! / (163)!, or 16! / 13!

Combination where order DOESN'T matter, repetition is not allowed (like a lottery)
n! / (r![nr]!)
n is number of things to choose from, you choose r

Combination where order DOESN'T matter, you can repeat (picking scoops of ice cream)
(n+r1)! / (r![n1]!)
n is number of things to choose from, you choose r of them

Probabilty that X AND Y both occur
Multiply probabilities together

Probability that independent event X OR event Y will occur
Add probabilities together

Probability of event X OR event Y, which can occur together
P(X) + P(Y)  P(X+Y)

