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k-permutations and k-combinations of setsLecture 2
(Brualdi Ch. 2.2, 2.3)
Mikhail Ivanov
Wednesday, January 27th
De�nition: Given a set S. Given integer k > 0. An k-permutation of S is a
sequence
a1,a2, . . . ,ak
where a1, a2, . . . , ak are distinct elements of S. By a permutation of S we
mean an n-permutation, where n = |S|.
Example: S = {a,b, c}.
The permutations of S are
The 2-permutations of S are
The 1-permutations of S are
The 0-permutation of S is
No k-permutations of S for k > 3.
Lecture 2 page 2 ,
Example: Given �nite set S, write n = |S|. For 0 6 k 6 n �nd the
number of k-permutations of S.
De�nition: For an integer n > 0 de�ne factorial of n:
n! = n · (n− 1) · (n− 2) · . . . · 2 · 1.
Example: At a party there are 8 women and 11 men. How many ways are
there to form 8 couples consisting of 1 man and 1 woman?
Lecture 2 page 3 ,
Example: Given set S of people |S| = n. For 1 6 k 6 n seat k people
from S around circular table. People care only about who seats left and right to
you. How many arrangements possible?
In summary:
Theorem: Given a �nite set S with |S| = n. For 1 6 k 6 n the number of
circular k-permutations of S is
P(n,k)
k=
n!
k · (n− k)!.
in particular, taking k = n, the number of circular permutations of S is
Lecture 2 page 4 ,
Example: What is the number of ways to order the 26 letters of the
alphabet such way that no two of the vowels occur consecutively?
Combinations of Sets
De�nition: Given a set S, |S| = n. For 0 6 k 6 n by k-combination (or
k-subset) of S we mean a subset of S with cardinality k.
Example: Find the number of k-combinations of S, |S| = n.
Lecture 2 page 5 ,
Convention: (n
k
)= if k < 0 or k > n.
Properties:(n
0
)=
(n
n
)=
(n
k
)=
Pascal triangle:
Lecture 2 page 6 ,
Theorem: For 1 6 k 6 n− 1(n
k
)=
(n− 1
k
)+
(n− 1
k− 1
)
Lecture 2 page 7 ,
Theorem: For an integer n > 0(n
0
)+
(n
1
)+ . . .+
(n
n
)= 2n.
Lecture 2 page 8 ,
Example: Consider 4× 6 grid of city streets: �nish if 6 blocks East and 4
blocks North from start. How many ways one have to go in 10 steps, each
North or East from start to �nish?