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The Addition & Multiplication Principles 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 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 10 30 = 70 Example • In how many ways can we draw a heart or a spade from an ordinary deck of cards? • A heart or an ace? • 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. The Product Rule • Let S1, S2, . . . , Sn be nonempty sets. • | S1 S2 . . . Sn | = | S1||S2| . . . |Sn |. b a (a,0) (a,1) (b,0) c (b,1) S1 = {a, b, c}; S2 = {0, 1} (c,0) (c,1) • 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? 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? • 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? 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 < 109 contain the digit 1? • The size of the universe is 109. • The number of nonnegative numbers < 109 that do not contain the digit 1 is 99. • This includes numbers like 000000004, which is 4. 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. 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!). 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. 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. • 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. • 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. 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. Characters • • • • • •