Permutations, combinations, and variations
1 Permutations
Permutations are arrangements of objects (with or without repetition), order does matter.
The number of permutations of nobjects, without repetition, is
Pn=Pn
n=n!.
The counting problem is the same as putting ndistinct balls into ndistinct boxes, or to count bijections
from a set of ndistinct elements to a set of ndistinct elements.
A permutation with repetition is an arrangement of objects, where some objects are repeated a
prescribed number of times. The number of permutations with repetitions of k1copies of 1,k2copies
of 2, .. . , krcopies of ris
Pk1,...,kr=(k1+· · · +kr)!
Qr
i=1 ki!
The counting problem is the same as putting k1+· · · +krdistinct balls into rdistinct boxes such
that box number ireceives kiballs. In other words we count onto functions from a set of k1+· · · +kr
distinct elements onto the set {1,2, . . . , r}, such that the preimage of the element ihas size ki.
2 Combinations
Combinations are selections of objects, with or without repetition, order does not matter.
The number of k-element combinations of nobjects, without repetition is
Cn,k =n
k=n!
k!(n−k)!.
The counting problem is the same as the number of ways of putting kidentical balls into ndistinct
boxes, such that each box receives at most one ball. It is also the number of one-to-one functions
from a set of kidentical elements into a set of ndistinct elements. It is also the number of k-element
subsets of an n-element set.
The number of k-element combinations of nobjects, with repetition is
Cn,k =Cn+k−1,k =n+k−1
k=n
k.
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