Permutation With Permutation With Repetition and Circular Repetition and Circular PermutationsPermutations Linear Permutations:Linear Permutations: arrangements arrangements

in a line: n!in a line: n! Permutations involving Permutations involving

Repetitions:Repetitions: n!/(p!q!) n!/(p!q!) How many thirteen-letter patterns How many thirteen-letter patterns

can be formed from the letters of can be formed from the letters of the word the word differentiatedifferentiate??

13!/(2!2!3!2!)13!/(2!2!3!2!) =129,729,600=129,729,600

Permutation With Permutation With Repetition and Circular Repetition and Circular PermutationsPermutations Circular Permutations:Circular Permutations: No beginning or end..... :No beginning or end..... : (n-1)! (n-1)!

CChildren on a Merry-Go-Roundhildren on a Merry-Go-Round Fixed point of Fixed point of

referencereference.....considered Linear : n! .....considered Linear : n! Seating around a round table with one person Seating around a round table with one person next to a computernext to a computer

If the circular permutation looks If the circular permutation looks the same when it is turned over, the same when it is turned over, such as a plain key ring, then the such as a plain key ring, then the number of permutations must be number of permutations must be divided by two.divided by two.

Permutation With Permutation With Repetition and Circular Repetition and Circular PermutationsPermutations During an activity at school, 10 children During an activity at school, 10 children

are asked to sit in a circleare asked to sit in a circle Is the arrangement of children a linear or circular Is the arrangement of children a linear or circular

permutation? Explain.permutation? Explain.

arrangement is a circular permutation since the arrangement is a circular permutation since the children sit in a circle and there is no reference point.children sit in a circle and there is no reference point.

There are ten children so the number of There are ten children so the number of arrangements can be described by (10 - arrangements can be described by (10 - 1)! or 9!1)! or 9!

9! = 9 9! = 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 or 1 or 362,880362,880

Permutation With Permutation With Repetition and Circular Repetition and Circular PermutationsPermutations

Permutation With Permutation With Repetition and Circular Repetition and Circular PermutationsPermutations

Permutation With Permutation With Repetition and Circular Repetition and Circular PermutationsPermutations

Permutation With Permutation With Repetition and Circular Repetition and Circular PermutationsPermutations