1. Permutation With Repetition and Circular Permutations
Linear Permutations: arrangements in a line: n!
Permutations involving Repetitions: n!/(p!q!)
How many thirteen-letter patterns can be formed from the letters of the word differentiate?
13!/(2!2!3!2!)
=129,729,600

2. Permutation With Repetition and Circular Permutations
No beginning or end..... : (n-1)! Children on a Merry-Go-Round
Fixed point of reference.....considered Linear : n! Seating around a round table with one person next to a computer
If the circular permutation looks the same when it is turned over, such as a plain key ring, then the number of permutations must be divided by two.

3. Permutation With Repetition and Circular Permutations
During an activity at school, 10 children are asked to sit in a circle
Is the arrangement of children a linear or circular permutation? Explain.
arrangement is a circular permutation since the children sit in a circle and there is no reference point.
There are ten children so the number of arrangements can be described by (10 - 1)! or 9! 9! = 9  8  7  6  5  4  3  2  1 or 362,880

4. Permutation With Repetition and Circular Permutations

5. Permutation With Repetition and Circular Permutations

6. Permutation With Repetition and Circular Permutations

7. Permutation With Repetition and Circular Permutations