Counting

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Section 3-6

• Counting

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FUNDAMENTAL COUNTING RULE
For a sequence of two events in which the first
event can occur m ways and the second event can
occur n ways, the events together can occur a
total of m n ways. This generalizes to more
than two events.
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EXAMPLES

1. How many two letter words can be formed if the
   first letter is one of the vowels a, e, i, o, u
   and the second letter is a consonant?
2. OVER FIFTY TYPES OF PIZZA! says the sign as you
   drive up. Inside you discover only the choices
   onions, peppers, mushrooms, sausage, anchovies,
   and meatballs. There are also 3 different types
   of crust and 4 types of cheese. Did the
   advertisement lie?
3. Janet has five different books that she wishes to
   arrange on her desk. How many different
   arrangements are possible?
4. Suppose Janet only wants to arrange three of her
   five books on her desk. How many ways can she do
   that?

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FACTORIALS
NOTE 0! is defined to be 1. That is, 0! 1
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FACTORIAL RULE
A collection of n objects can be arranged in
order n! different ways.
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PERMUTATIONS
A permutation is an ordered arrangement. A
permutation is sometimes called a sequence.
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PERMUTATION RULE(WHEN ITEMS ARE ALL DIFFERENT)
The number of permutations (or sequences) of r
items selected from n available items (without
replacement) is denoted by nPr and is given by
the formula
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PERMUATION RULE CONDITIONS

• We must have a total of n different items
  available. (This rule does not apply if some
  items are identical to others.)
• We must select r of the n items without
  replacement.
• We must consider rearrangements of the same items
  to be different sequences. (The arrangement ABC
  is the different from the arrangement CBA.)

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EXAMPLE
Suppose 8 people enter an event in a swim meet.
Assuming there are no ties, how many ways could
the gold, silver, and bronze prizes be awarded?
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PERMUTATION RULE(WHEN SOME ITEMS ARE IDENTICAL
TO OTHERS)
If there are n items with n1 alike, n2 alike, . .
. , nk alike, the number of permutations of all n
items is
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EXAMPLE
How many different ways can you rearrange the
letters of the word level?
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COMBINATIONS
A combination is a selection of objects without
regard to order.
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COMBINATIONS RULE
The number of combinations of r items selected
from n different items is denoted by nCr and is
given by the formula
NOTE Sometimes nCr is denoted by
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COMBINATIONS RULE CONDITIONS

• We must have a total of n different items
  available.
• We must select r of those items without
  replacement.
• We must consider rearrangements of the same items
  to be the same. (The combination ABC is the same
  as the combination CBA.)

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EXAMPLES

1. From a group of 30 employees, 3 are to be
   selected to be on a special committee. In how
   many different ways can the employees be
   selected?
2. If you play the New York regional lottery where
   six winning numbers are drawn from 1, 2, 3, . . .
   , 31, what is the probability that you are a
   winner?
3. Exercise 34 on page 183, 15-18 page 188.

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EXAMPLE
Suppose you are dealt two cards from a
well-shuffled deck. What is the probability of
being dealt an ace and a heart?
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PERMUTATIONS VERSUS COMBINATIONS
When different orderings of the same items are to
be counted separately, we have a permutation
problem, but when different orderings are not to
be counted separately, we have a combination
problem.