Title: Combinatorics
1Combinatorics
- CS311
- Western Washington University
2What is Combinatorics?
The mathematics of combinations. What is a
combination? An ordered sequence. To start,
lets look at some elementary counting
techniques for both ordered and unordered sets
of data.
3Permutation
How many different ways can we arrange
the elements of a set? Order is very
important. If set S has n members, then there
are n choices for the first element, n-1
choices for the second element, etc. Therefore
there are n(n-1)(n-2)1 possible arrangements
n! What if we want to count the number of
permutations of r elements chosen from an n
element set? n(n-1)(n-r1) This is also
expressed as P(n, r) n!/(n-r)! If Sa, b, c,
d, how many 2 element permutations exist? P(4,2)
4!/(4-2)!
4Permutations and Bags
Let B be an n-element bag with k
distinct elements. Let m1, m2, , mk denote the
number of occurrences of each element. The
number of permutations of the n elements of B
is n! / m1!m2!mk! Using this
information Build a code to represent each of
29 distinct objects with a binary string having
the same minimal length n, where each string has
the same number of 0s and 1s. n!/k!k! gt 29
where k n/2. Can be solved by trial and error.
5Combinations
Combinations are used for counting when order
doesnt matter, I.e. when counting the number of
subsets. Combinations are C(n,r) P(n,r)/r!
n!/r!(n-r)! Where did this formula come
from? Start by counting the number of r element
permutations of n elements. In this count, we
have included r! distinct r-element
permutations. We need to remove these redundant
permutations from the count.
6Naturally Occurring Combinations
The binomial coefficient is an example of a
Naturally occurring combination. (ab)4 a4
4a3b 6a2b2 4ab3 b4 C(4,0)a4 C(4,1)a3b
C(4,2)a2b2 C(4,3)ab3 C(4,4)b4 Pascals
triangle contains the binomial Coefficients for
(ab)n.
7Bag Combinations
We may have the need to count bags of
things rather than sets. The number of k-element
bags whose distinct elements are chosen from an
n-element set is C(n k 1, k) Example In
how many ways can five people be selected from a
collection of democrats, republicans, and
independents? C(3 5 1, 5)
8Probability Distributions
A probability distribution p on S is a
function PS -gt 0,1 Such that p(x1) p(x2)
p(xn) 1 For a coin toss, the 4 possible
outcomes are HH, TT, HT, TH , and p(HH)
p(TT) p(HT) p(TH) ¼ The probability of any
event E in S is denoted P(E) ?
p(x). Example Let E be the event that at least
1 coin in a toss Is a tail. Then E TT, HT, TH
and P(E) ¼ ¼ ¼ ¾
9The Birthday Problem
Given n people in a room, what is the
probability that at least 2 of the people
have the same birthday(month and day)? Assume
365 days in the year, so there are 365n possible
n-tuples of birthdays for n people. Also assume
that birthdays are evenly distributed. p(ltb1,
b2, , bngt) 1 /365n E is the subset of S
consisting of all n-tuples that contain 2 or
more equal entries. We want to know what is
P(E). We can solve this using the complement E
S E. This is the case where no 2 of the n
people have the same birthday. The
probability that we want is P(E) 1 P(E).
10P(E) P(365, n) / 365n Thus, P(E) 1
P(365,n) / 365n