1. 13.2 Permutations with Repetitions & Circular Permutations
Solve problems involving permutations with repetitions.
Solve problems involving circular permutations

2. Permutations with repetitions
Ex 1) How many 5-letter patterns can be formed from the letters of the word graph?
Ex 2) How many 5-letter patterns can be formed from the letters of the word jello?
Permutations with repetitions
The number of permutations of n objects of which p are alike and q are alike is 𝑛! 𝑝!𝑞!
3) How many 11-letter patterns can be formed from the letters of the word Mississippi?

3. Circular Permutations
If n objects are arranged in a circle, then there are 𝑛! 𝑛 𝑜𝑟 𝑛−1 ! permutations of the n objects around the circle.
Linear:
Circular:
Circular with a fixed point:

4. Examples.
How many different ways can the letters of each word be arranged?
1) Kangaroo 2) classical
3) In how many ways can 2 red lights, 4 yellow lights, 5 blue lights, 1 green light, and 2 pink lights be arranged on a string of lights?
Determine whether each arrangement of objects is linear or circular permutation. Then determine the number of arrangements.
4) 11 football players in a huddle
5) 8 jewels on a necklace
6) 5 beads on a string, in relation to a knot in the string.

5. 7) Eight people are to be seated at a table
7) Eight people are to be seated at a table. How many ways can the people be arranged?
8) Eight people are to be seated at a table where one person is seated next to a window. How many ways can the people be arranged relative to the window?
9) Morse code is a system of dots, dashes, and spaces that telegraphers in the US once used to send messages by wire. How many different arrangements are there of 5 dots, 2 dashes, and 2 spaces?