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## Section 15-4

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**Section 15-4**Permutations with Repetition; Circular Permutations**Activity**• Complete activity on p. 583 • Write down all possible arrangements of the letters of the word MOP. • Write down all possible arrangements of the letters of the word MOM. • What accounts for the fact that part (b) gives fewer permutations than part (a)?**Activity**• As the activity shows, fewer permutations result when some of the objects being rearranged are the same.**The Number of Permutations of Things Not All Different**• Let S be a set of n elements of k different types. Let = the number of elements of type 1, = the number of elements of type 2, …, = number of elements of type k. Then the number of distinguishable permutations of the n elements is:**Circular Permutations**• So far we have discussed only linear permutations, but permutations may also be circular (or cyclic). Discuss example on the bottom of p. 584 These circular permutations are the same, because in each one, A is to the right of B, who is to the right of C, who is to the right of D. These linear permutations are different. ABCD DABC CDAB BCDA