Transcript Permutations and Combinations

Discrete Structures, Fall 2011
Permutation vs Combination
Permutations
Combinations
 Ordering of elements from a
 Collection of element from a
set
 Sequence does matter
 1 2 3 is not the same as 3 2 1
set
 Order does not matter
 1 2 3 is the same as 3 2 1
Permutation Example
 How many ways can we select 3 students from a group of 5 students to
form a line to wait to order lunch?
 Students = S = { 1 2 3 4 5 }
 Possible permutations would be:




123
345
213
542
 Solution: notice that order matters
 There are 5 ways to select the first student
 Once that selection is made, there are 4 ways to select the 2nd student
 Then there are 3 ways to select the 3rd student.
 5*4*3 = 60. There are sixty different permutations
 Alternatively, to arrange all 5 students in a line, then there are 5 * 4 * 3 *
2 * 1 = 120 permutations.
Permutation Definition
 A Permutation is an ordering of the objects within a
distinct set.
 An arrangement of a subset of the original set is called
an r-Permutation, where the “r” is the number of
elements in the subset
 In the previous example, we first calculated the 3permutation and then calculated the full permutation.
 Consider the set S = { 1, 2, 3 }
 The arrangement 2 1 3 is a permutation of S, while 2 1 is a
2-permutation of S
How many permutations of a set
exist?
 For a given set containing n elements, we can calculate the
number of possible full permutations by considering the number
of options we have at each assignment
 # of permutations = (n)(n-1)(n-2)(n-3)…(2)(1) = n!
 n! is called “factorial of n” and is n multiplied by every number
between n and 0.
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5! = 5*4*3*2*1
0! = 1 (special case)
 For an r-permutation, we have:
 # of permutations = (n)(n-1)(n-2)(n-3)…(n-r+1)
 Introduce a new notation for the r-permutation:


P(n,r) = (n)(n-1)(n-2)(n-3)…(n-r+1)
P(n,r) = n!/(n-r)!

where n is the number of elements in the set, and r is the number of
elements in the permutation
Example
 How many different ways are there to select a first-
prize winner, a second-prize winner and a third-prize
winner from 100 different people who entered a
contest?
 This is a 3-permutation
 P(100,3) = 100 * 99 *98 = 970,200
Combinations
 How many different committees of three students can be
formed by a group of 4 students?
 S={1234}
 Solution:
 We need to find the number of subsets with three elements.
 This is the same as eliminating 1 number.
 There are 4 subsets, or combinations.
 This is an example of finding 3-Combinations of a set S.
 The number of r distinct combinations of the set of size n is
denoted C(n,r)
 To compute C(n,r), use the formula n!/(r!*(n-r)!)
Combination Example
 How many poker hands of 5 cards can be dealt from a
deck of 52 cards?
 Order does not matter, so this is a combination
problem.
 C(52,5) = 52!/5!47!
 = 52*51*50*49*48/(5*4*3*2*1) = 2,598,960
Class problems:
 List the permutations of {a b c}
 How many different permutations are there of { a b c d
e f g }?
 Let S = {1 2 3 4 5}
 List the 3-permutations of S
 List the 3-combinations of S
 Find C(5,3)
 Are there more n-combinations or n-permutations of a
set?