The Addition

1
The Addition Multiplication Principles
2
The Addition Principle

• If S1, S2, . . . , Sn is a partition of S,
• then S S1 S2 . . . Sn.
• Example
• Let S1, S2 be a partition of S S1 S2
  40.
• Then S 80.

40
40

80
3
When A B are not Disjoint

• A - B, A ? B, B - A partition A ? B
• Every element is in exactly 1 part.
• Let A-B 30, A ? B 10, B-A 30.
• A ? B A - B A ? B B - A 70.

30
30
10
70
4
Example

• In how many ways can we draw a heart or a spade
  from an ordinary deck of cards?
• A heart or an ace?

5

• In how many ways can we get a sum of 4 or 8 when
  2 distinguishable dice (say 1 is red, 1 blue) are
  rolled?
• Model distinguishable as an ordered pair.
• What about when the dice are indistinguishable?
• Model indistinguishable as a set.

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The Product Rule

• Let S1, S2, . . . , Sn be nonempty sets.
• S1 ? S2 ? . . . ? Sn S1?S2? . . . ?
  Sn .

b
a
c
(a,0)
(a,1)
(c,0)
(c,1)
(b,0)
(b,1)
S1 a, b, c S2 0, 1
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• Think of the product rule as creating a composite
  object in stages S1, S2, . . . , Sn .
• Then there are S1?S2? . . . ? Sn
  different composite objects.
• If 2 distinct dice are rolled, how many outcomes
  are there?
• If 100 distinct dice are rolled, how many
  outcomes are there?

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Example

• Suppose the CCS tee shirt comes in 3 colors, and
  4 sizes. How many different kinds of CCS tee
  shirts are there?
• How many 3-digit numbers can be formed from the
  digits 1, 2, 3, 4, 5, 6, 7, 8, 9?
• How many 3-digit numbers can be formed from the
  above set when no digit can be repeated?

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• How many license plates can be formed from 3
  letters followed by 4 digits?
• From 1, 2, or 3 letters, followed in each case by
  4 digits?
• From 1, 2, or 3 letters, followed in each case by
  4 digits, when the 4 digits, interpreted as a
  number, is even?

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Indirect Counting

• Count the elements of a set by computing the size
  of its complement subtracting from size of the
  universe.
• How many nonnegative numbers lt 109 contain the
  digit 1?
• The size of the universe is 109.
• The number of nonnegative numbers lt 109 that do
  not contain the digit 1 is 99.
• This includes numbers like 000000004, which is 4.

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Example

• We draw a card from a deck replace it before
  the next draw. In how many ways can 10 cards be
  drawn so that the 10th card matches at least 1 of
  the previous draws?
• There are 5210unrestricted 10 card draws.
• There are (52)(51)9 ways to draw 10 cards where
  the 10th card does not match any previous
• 1st pick the 10th card pick the other 9 from the
  other 51 cards.

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Example 2

• How many ways can 8 students be seated in a row
  so that a certain pair are not adjacent?
• There are 8! seatings without restriction.
• The number of ways where the pair do sit next to
  each other is (7)(2)6! (2)7!
• Pick the position of the left seat of the pair
  (7)
• Pick the order of the pair (2)
• Order the other 6 people in the other 6 seats
  (6!).

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One-to-one Correspondence

• 1) Note that the number of solutions to one
  problem is in 1-to-1 correspondence with those of
  another problem.
• 2) Count the number of solutions in the other
  problem (which presumably is easier).
• Example To count the number of cows in a field,
  simply count the number of cow legs in the field
  and divide by 4.

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Ok, a real example

• Suppose there are 101 players in a single
  elimination tennis tournament.
• In such a tournament, if a player loses a match,
  he is eliminated (i.e., shot).
• In every match, someone loses (no ties).
• The tournament proceeds in rounds.
• In round 1, there are 50 matches, someone gets
  a bye.

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• In round 2, there are 25 matches, someone gets
  a bye.
• In round 3, there are 13 matches.
• In round 4, there are 6 matches, and someone gets
  a bye.
• In round 5, there are 2 matches.
• In round 6, there is the final match.

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• If there are 10,975 people in the tournament, how
  many matches are needed?
• Observe that there is a 1-to-1 correspondence
  with matches and losers.
• A match eliminates 1 person from the tournament.
• Thus, 10,975 - 1 matches are needed.

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Attacking a problem

• Devising an overall counting strategy usually is
  the hardest part.
• If it is unclear how to proceed, get concrete
• Start to enumerate the possible outcomes - this
  usually leads to some insight as to the structure
  of the problem.
• Try a special case. For example, if the problem
  is in terms of a parameter, n, try to solve it
  for n 2 then 3 then 4. Look for a pattern.

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Characters

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