Product Rule - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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Discrete Mathematics

CS 2610

October 28, 2008

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Counting

Part of combinatorics, the study of arrangements

of objects. (Sets, sequences, sebsets, etc.)

Counting relies on two important, but simple principles: the Product Rule and Sum Rule

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Counting

Sum Rule  If a procedure can be broken down into a choice of one of I distinct, exhaustive tasks and there are n (^1) ways to accomplish the first, n 2 ways to accomplish the second, etc., then there are n 1 + n 2 + … + n (^) I ways to accomplish the procedure.

Examples:  How many subsets of a set of size 3?  Let si be the number of subsets of size I  So the answer is s 0 + s 1 + s 2 + s 3 = 1 + 3 + 3 + 1 = 8  (You can derive each si with the product rule.)

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Counting

Some problems are straightforward applications of one of these rules. Some problems involve using both rules.

Example: How many 6-8 character passwords containing at least one digit?

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Counting

Subtraction Principle : When using the sum rule, if the subtasks are not distinct, the number of common elements must be subtracted from the sum.

If A and B are sets, then |A ∪ B| = |A| + |B| - |A ∩ B| Don’t double count!!!

Example: How many bit strings of length 8 begin with 1 or end with 00?  2 7 + 2 6 - 2 5 = 128 + 64 -32 = 160.

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Counting

In some complex counting problems, decision trees will be useful. This is most common when the number of steps necessary depends on the choices made.

Example: How many ways are there to have a best- of-3 tournament between two cribbage players?

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The Pigeonhole Principle

Examples:  Among 100 people, at least how many must have been born in the same month?

 How many 3-digit area codes are necessary to guarantee unique 10-digit phone numbers for a state with 25 million residents?

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Permutations and Combinations

A permutation of a set of distinct objects is an ordered arrangement (list) of these objects.

An r-permutation of a set of distinct objects is an

ordered arrangement of a subset of size r.

The number of r-permutations of a set with n

elements is given by the product rule P(n,r) = n ⋅ (n-1) ⋅ … ⋅ (n-r+1), or

P(n,r) = n! / (n-r)!, for 0 ≤ r ≤ n

Example: How many ways to award medals in a race with 8 people?

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Permutations and Combinations

Interesting fact about combinations:  for 0 ≤ r ≤ n, C(n,r) = C(n,n-r)

Why? Because when you select a subset, you actually select two subsets at once- those that are chosen and those that aren’t!

Another example:  How many ways to select a 22-person All-SEC football team consisting of 11 UGA starters and 1 starter from every other SEC team?