Counting Problems and Principles

Counting Problems and Principles

This article discusses different methods for solving counting problems, including the multiplication principle, permutations, and combinations. It covers how to apply these principles to various scenarios, such as when dealing with non-distinct objects

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PowerPoint presentation about 'Counting Problems and Principles'. This presentation describes the topic on This article discusses different methods for solving counting problems, including the multiplication principle, permutations, and combinations. It covers how to apply these principles to various scenarios, such as when dealing with non-distinct objects. The key topics included in this slideshow are . Download this presentation absolutely free.

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Slide1Permutations and CombinationsSolve Counting Problems Using Multiplication Principle Solve Counting Problems Using Permutations Solve Counting Problems Using Combinations Solve Counting Problems Using Permutations involving non distinct objects

Slide2Multiplication Principle of CountingIf a task consists of a sequence of choices in which there are p selections for the first choice, q selections for the second choice and r selections for the third choice and so on then the task of making these selections can be done in p ∙ q ∙ r ∙ ……..

Slide3QuestionThe fixed price dinner at a restaurant provides the following choices Appetizer: Soup or Salad Entrée:  Chicken, Beef, Fish or Pork Dessert:  Ice Cream or Cheesecake How many different meals can be ordered?

Slide4PermutationA permutation is an ordered arrangement of r objects chosen from n objects

Slide5Permutations: Distinct objects withrepetition The number of ordered arrangements of r objects chosen from n objects in which the n objects are distinct and repetition is allowed is equal to n r . The symbol P(n,r) represents the number of ordered arrangements of r objects chosen from n distinct objects where r ≤ n and repetition is not allowed.

Slide6Permutations of r objects chosen fromn distinct objects without repetition The number of arrangements of n objects using r  ≤ n of them in which 1. The n objects are distinct 2. Once an object is used it can not be used again 3. Order is important Is given by the formula  

Slide7ExampleThe International Airline Transportation Association assigns three letter codes to represent airport locations.  For example the airport code for Ft Lauderdale, Florida is FLL. Notice that repetition is allowed in forming this code.  How many airport codes are possible?

Slide8ExampleSuppose that we wish to establish a three letter code using any of the 26 uppercase letters of the alphabet, but we require that no letter be used more than once.  How many different three letter codes are there?

Slide9Lining up peopleIn how many ways can 5 people be lined up?

Slide10Birthday problemAll we know about Shannon, Patrick and Ryan is that they have different birthdays.  If we listed all the possible ways this could occur how many would there be?  Assume there are 365 days in a year

Slide11CombinationsA combination is an arrangement, without regard to order, of r objects selected from n distinct objects without repetition, where r  ≤n.  The symbol C(n,r) represents the number of combinations of n distinct objects using r of them.

Slide12Forming CommitteesHow many different committees of 3 people can be formed from a pool of 7 people?

Slide13Forming Committees 2In how many ways can a committee consisting of 2 faculty members and 3 students be formed if 6 faculty members and 10 students are eligible to serve on the committee?

Slide14Forming different wordsHow many different words (real or imaginary) can be formed using all the letters in the word REARRANGE?

Slide15Number of combinations of n distinctobjects taken r at a time The number of arrangements of n objects using r ≤n of them, in which 1. Th n objects are distinct 2. Once an object is used, it cannot be repeated 3. Order is not important is given by the formula  

Slide16Permutations involving n objects thatare not distinct The number of permutations of n objects of which n 1  are of one kind and n 2  are of a second kind…….and n k  are of the k th  kind is given by

Slide17QuestionThe student relations committee of a college consists of 2 administrators 3 faculty members and 5 students. Four administrators, 8 faculty members, and 20 students are eligible to serve. How many different committees are possible?

Slide18QuestionHow many different 9 letter words (real and imaginary) can be formed from the letters in the word ECONOMICS?

Slide19QuestionHow many different 11 letter words (real or imaginary) can be formed from the letters in the word MATHEMATICS?