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?