Download Permutations with Repetitions and Circular

Permutations with Repetitions and Circular Permutations
Permutation with Repetition: The number of permutations of n objects of which p are indistinguishable and q are
indistinguishable is n!/(p!q!)
Example: How many 6-letter patterns can be formed from the letters of the word BOTTOM?
Answer: 6!/(2!2!)=180
Determine how many different letter patterns can be formed.
1. CANADA
6!/3!=120
2. MEMBERS
7!/(2!2!)=1,260
3. ILLINOUIS
9!/(2!3!)=30,240
4. ANNUALLY
8!/(2!2!2!)=5,040
5. QUOUNNTAGGRA
12!/(2!2!2!2!)=29,937,600
6. STEREO
6!/2!=360
Solve each problem.
1. How many ways can 3 identical pens and 5 identical pencils be given to 8 students?
8!/(3!5!)=56
2. How many 5-digit numbers can be made using the digits from 76,627?
5!/(2!2!)=30
3. Eight cards, all the same size are arranged in a row. Six of the cards are black and two of the cards are white.
Cards of the same color are indistinguishable. How many possible arrangements are possible?
8!/(6!2!)=28
Circular Permutations
Circular Permutations: If n objects are arranged in a circle (with NO fixed point), then there are (n-1)!
permutations of the n objects around the circle.
Example: A supermarket offers party trays that are made up of six different deli items. How many
different ways can these items be arranged on the tray if each item must fit in its own section?
Answer: (6-1)!=5!=120
Reflections: If an arrangement can be flipped over, then the two arrangements are reflections of each other.
There are only half as many arrangements when reflection is possible.
Example: Four keys are placed on a key ring (no endpoint). How many different arrangements are possible?
Answer: (4-1)!/2=3
Complete: Under what condition is a circular permutation treated as linear?
When there is a fixed point (example-clasp)
Solve each problem.
1. How many ways can six campers be arranged around a campfire?
(6-1)!=120
2. How many ways can 8 charms be arranged on a bracelet that has NO CLASP?
(8-1)!/2=2,520
3. How many ways can 8 charms be arranged on a bracelet that HAS a clasp?
8!/2=20,160
4. How many ways can 11 players form a football huddle?
(11-1)!=3,628,800