MATH 2311 Section 2.1. Counting Techniques Combinatorics is the study of the number of ways a set of objects can be arranged, combined, or chosen; or.

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MATH 2311 Section 2.1.

Presentation transcript:

MATH 2311 Section 2.1

Counting Techniques Combinatorics is the study of the number of ways a set of objects can be arranged, combined, or chosen; or the number of ways a succession of events can occur. Each result is called an outcome. An event is a subset of outcomes. When several events occur together, we have a compound event.

The Fundamental Counting Principle The Fundamental Counting Principle states that the total number of a ways a compound event may occur is n 1  n 2  n 3 ....  n i where n 1 represents the number of ways the first event may occur, n 2 represents the number of ways the second event may occur, and so on.

Example: How many ways can you create a pizza choosing a meat and two veggies if you have 3 choices of meats and 4 choices for veggies?

Permutations A permutation of a set of n objects is an ordered arrangement of the objects. n P n  n(n  1)(n  2)....3  2  1  n! All objects are placed in order Some of the objects are placed in order

Examples: How many ways can six people be seated in a row? In how many ways can 3 of the six symbols, be arranged?

With Repetition When we allow repeated values, The number of orderings of n objects taken r at a time, with repetition is n r. Example: In how many ways can you write 4 letters on a tag using each of the letters C O U G A R with repetition?

Duplicate Objects The number of permutations, P, of n objects taken n at a time with r objects alike, s of another kind alike, and t of another kind alike is

Example: How many different words (they do not have to be real words) can be formed from the letters in the word MISSISSIPPI?

Circular Permutations The number of circular permutations of n objects is (n – 1)! Example: In how many ways can 12 people be seated around a circular table?

Combinations: A combination gives the number of ways of picking r unordered outcomes from n possibilities. The number of combinations of a set of n objects taken r at a time is

Example: How many ways can a committee of 5 be chosen from a group of 12 people?

Permutation or Combination: How many ways can 1 st, 2 nd, and 3 rd place ribbons be awarded when there are 15 contestants? How many ways can you be dealt a poker hand of 7 cards from a standard deck of 52? How many ways can a class President, Vice President, Secretary, Treasurer, and Historian be selected from a class of 500?