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ZingPath: Permutations and Combinations

Circular Permutation Problems

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Permutations and Combinations

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Lesson Focus

Circular Permutation Problems

Geometry

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Solve problems involving circular permutations using the appropriate formula in round table and bracelet - necklace examples.

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Now You Know

After completing this tutorial, you will be able to complete the following:

  • Apply the formula <img src../../../tutorials/images/teacherguides/US840408PS_1.png\"width=\"58\"height=\"25\"alt=\" border=0\"> to solve circular permutation problems.
  • Apply the formula <img src../../../tutorials/images/teacherguides/US840408PS_2.png\"width=\"59\"height=\"47\"alt=\" border=0\"> to solve circular permutation problems.
  • Solve problems involving circular permutations.

Everything You'll Have Covered

Given n items, you can determine the number of circular configurations given certain conditions. Some sample questions from the Activity Object follow:

  • How many ways can n people can be seated at a round table?
  • How many ways can n beads can be arranged on a bracelet?

Rotation yields no new arrangement

Rotating the people at a table or beads on a bracelet does not increase the number of alternative arrangements. Rotation only moves an existing arrangement. If everyone at a round table moves one seat to the left, the individuals are still seated in the same arrangement.

Determining possible arrangements

A relative position is a point defined with reference to another position, either fixed or moving. To determine the number of arrangements, a fixed position is set. The other elements are placed in relative positions to that fixed position. In the Activity Object, Section 1, after choosing one person to sit in a fixed position, we learn:

  • Possible choices for seat two = 3
  • Possible choices for seat three = 2
  • Possible choices for seat four = 1

So, the total number of different seating arrangements is possible arrangements.

Counting principle: multiplication

The solution to the question "How many ways can 4 people can be seated at a round table?" involves the counting principle of multiplication. The definition is:

A method used to calculate all of the possible outcomes of a given number of events.

The formula is:

where is the number of possible outcomes of event 1, is the number of possible outcomes of event 2, etc., and is the product of all of these event possible outcomes multiplied together.

Tutorial Details

Approximate Time 20 Minutes
Pre-requisite Concepts counting principle, factorial
Course Geometry
Type of Tutorial Problem Solving & Reasoning
Key Vocabulary circular permutations, counting principle, factorial