May 10, 2011
Section 9.1
Basic Combinatorics
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Multiplication Principle of Counting
If a procedure P has a sequence of stages
S1, S2, ..., Sn and if
S1 can occur in r1 ways,
S2 can occur in r2 ways,
Sn can occur in rn ways,
then the number of ways that P can occur is
r1 ⋅ r2⋅ ... ⋅rn.
How many license plates can be made if
the first two characters are digits 1-9, the
next three characters are letters, and the
final character is a digit or letter?
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May 10, 2011
The number of ways that a set of n objects
can be arranged in order is a permutation.
In a permutation order matters.
Permutation Counting Formula
The number of permutations of n objects
taken r at a time is nPr and is given by
nPr
How many ways can a 1st, 2nd, and 3rd
prize be awarded to 22 students who take
a math contest?
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How many ways can 3 equal prizes be
awarded to the same students?
If a group of n objects are taken r at a time then
order no longer matters. These unordered
selections are called combinations.
Combination Counting Formula
The number of combinations of n objects
taken r at a time is nCr and is given by
nCr
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Examples:
How many distinguishable 11-letter "words"
can be formed using the letters in
CHATTANOOGA?
Chattanooga has 11 letters: 3 A's, 1 C, 1 G, 1 H, 1 N, 2 O's, & 2 T's.
There are
"words".
Evaluate each without a calculator.
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A coin is tossed 20 times and the heads and
tails sequence is recorded. From among the
sequences of heads and tails, how many
have exactly seven heads?
Does order matter ?
What is an equivalent question?
From among the sequences
of heads and tails, how many
have exactly thirteen tails?
A new car customer has to choose from among
3 models, each of which comes in 4 exterior
colors, 3 interior colors, and with any
combination of up to 6 optional accessories.
How many essentially different ways can the
customer order the car?
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