The Addition & Multiplication Principles

The Addition Principle If {S 1, S 2,..., S n } is a partition of S, then |S| = |S 1 | + |S 2 | |S n |. Example: Let {S 1, S 2 } be a partition of S; |S 1 | = |S 2 | = 40. Then |S| = = 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 10

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 S 1, S 2,..., S n be nonempty sets. | S 1  S 2 ...  S n | = | S 1 |  |S 2 | ...  |S n |. a b c (a,0)(a,1) (b,0)(b,1) (c,0)(c,1) S 1 = {a, b, c}; S 2 = {0, 1}

Think of the product rule as creating a composite object in stages S 1, S 2,..., S n. Then there are | S 1 |  |S 2 | ...  |S n | 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 < 10 9 contain the digit 1? The size of the universe is The number of nonnegative numbers < 10 9 that do not contain the digit 1 is 9 9. This includes numbers like , 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 unrestricted 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, 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.

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