Systems can permute things in sequences of positions. Systems can have things in sequences of positions but sequence does not matter or some things are the same, so some permutations are the same {combination, mathematics}. Combinations are number of different permutations. Number of combinations is less than or equal to number of permutations.
permutations
N positions and N things have number of permutations N! = N * (N - 1) * (N - 2) * ... * 1. N things and R positions, with R < N, have number of permutations N! / (N - R)! = N * (N - 1) * (N - 2) * ... * (N - (R - 1)). N things and R positions, with N <= R, have number of permutations N! = N * (N - 1) * (N - 2) * ... * 1.
combinations
Number of combinations C equals number of permutations P divided by factorial of number of positions R: C = P / R!. Number of combinations of n things taken r at a time is n! / r!.
Number of combinations of n + 1 things taken r at a time equals number of combinations of n things taken r - 1 at a time, plus number of combinations of n things taken r at a time: (n + 1)! / r! = n! / (r - 1)! + n! / r!.
Number of combinations of n things taken n - r at a time equals number of combinations of n things taken r at a time: n! / (n - r)! = n! / r!.
Number of combinations of n things taken r + 1 at a time equals number of combinations of n things taken r at a time, times quantity (n - r) divided by (r + 1): n! / (r + 1)! = (n! / r!) * (n - r) / (r + 1).
binomial
Systems can have things that have states. Two things, each with possible states a and b, have four permutations and three combinations: 1 a*a, 2 a*b, and 1 b*b. Three things with states a and b have eight permutations and four combinations: 1 a*a*a, 3 a*a*b, 3 a*b*b, or 1 b*b*b, and so on.
Combination-term coefficients derive from binomial powers. (a + b)^2 = a^2 + 2*a*b + b^2, (a + b)^3 = a^3 + 3*a^2*b + 3*a*b^2 + b^3, (a + b)^4 = a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4, and so on.
First-event outcome can have no affect on second-event outcome {independent event}. For independent events, to find probability of outcome series, multiply event probabilities: P = p1 * p2 * p3 * ... * pN, where pi are probabilities, and N is event number.
Event outcome can mean that other outcomes cannot happen {disjoint events} {mutually exclusive events}|. For equally probable outcomes, outcome probability is 1/N, where N is number of mutually exclusive outcomes. To find probability of at least one of mutually exclusive outcomes, add all probabilities: P = p1 + p2 + p3 + ... + pN, where pi are probabilities, and N is outcome number.
Systems can have things that can be at sequence positions, or a thing can have a succession of states {permutation}| {arrangement}.
states
Number P of permutations for succession of states is product of number N of states raised to power of number R of events: N^R.
things
For sequences, number of permutations is product of number N of things times number of things minus one, and so on, until number R of sequence positions: N * (N - 1) * (N - 2) * ... * (N - R + 1). Using an outcome makes it unavailable for succeeding events.
Event series {stochastic process}| can have different results with different probabilities. Events can be independent {Bernoulli sequence}. Events can depend on preceding events in a Markov process.
Graphs {transition graph} can show event orders needed to reach all events.
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Date Modified: 2022.0225