Circular permutations and distinguishable permutations are discussed. Circular permutations refer to arranging objects in a fixed circle where order does not matter. The number of ways to seat 3 people at a circular table is 2 and the number of ways to seat 5 people is 4. Distinguishable permutations involve permutations with identical elements. The number of distinguishable permutations of letters in the words "basketball" and "heterogeneous" are calculated as examples. Rhoda can arrange 10 plants along her sidewalk consisting of 3 roses, 4 daffodils, and 3 lilies in 5040 distinguishable ways.

1. Circular permutaion
•It is the total number of ways
in which n distinct objects
can be arranged in a fixed
circle.
• Formula:

2. Example
•In how many ways can
three people be seated
around a circular table?

3. Example
•In how many ways can 5
people be seated around
a circular table?

4. DISTINGUISHABLE PERMUTATION
•It is a permutation with
identical element
•Formula:

5. Example
•Find the number of
distinguishable permutations
of the letters in each word.
•a. basketball
•b. heterogeneous

6. Example
• Rhoda wants to plant 10 plants
along the sidewalk in her frontyard.
She has 3 roses, 4 daffodils, and 3
lilies. In how many distinguishable
ways can the plants be arranged?