Combinatorics

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Combinatorics

• CS311
• Western Washington University

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What 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.
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Permutation
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)!
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Permutations 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.
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Combinations
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.
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Naturally 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.
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Bag 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)
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Probability 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
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The 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).
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P(E) P(365, n) / 365n Thus, P(E) 1
P(365,n) / 365n