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.
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Date Modified: 2022.0224