What is distinguishable permutations?

Question

It really didnt show me now kinda of defention for it on da internet

Answer 1

"Distinguishable permutations" are the usual kind of permutations: each of the things you are arranging is different.

Example: Alice, Bob, and Charlie run in a race. In how many different ways can they finish? Answer: 3P3 = 3! = 3x2x1. (They are
(Alice, Bob, Charlie),
(Alice, Charlie, Bob),
(Bob, Alice, Charlie),
(Bob, Charlie, Alice),
(Charlie, Alice, Bob), and
(Charlie, Bob, Alice)).

There are cases, however, where one or more of the things that you are arranging is the same, therefore "indistinguishable".

Example: In how many different ways can you arrange the letters of the word EGG? Answer: 3P3/2P2 = 3!/2! = 3. (They are EGG, GEG, and GGE.)

Answer 2

Hi,

A permutation is selecting things where their order is important, meaning that picking 1-2-3 is different from picking 3-1-2. The most common kind of problem with distinguishable permutations is when you rearrange the letters in a word. For example, suppose I asked you how many different ways could you rearrange the letters in the word "coin". The answer would be that you could put any of 4 letters in the first position, any of the remaining 3 letters in the second position, either of the last 2 letters in the third position and the last letter in the last position. That means there would be 4 X 3 X 2 X 1 = 24 ways to rearrange the letters in the word coin. Note that 4 X 3 X 2 X 1 = 4!, which is called 4 factorial and can be worked out on many calculators.

But, if a word has repeating letters, switching the letter "t" with another letter "t" would not make a distinguishable difference, they would both look the same. To find how many distinguishable permutations you could make for the word "rearrange", You would start by saying that since there are 9 letters in the word, you could make arrangements in 9! ways, if all the letters were different. BUT, because you have letters that repeat, all of those permutations would not be "distinguishable" from one another. To eliminate those indistinguishable permutations, you must divide the 9! permutations by a factorial telling the number of times each repeating letter occurs. So for "rearrange", you would have to do 9! divided by 3! for the letter r multiplied by 2! for the letter e times 2! for the repeating letter a. So your answer would be:

_____9!___
3! 2! 2!
This equals 15,120 distinguishable ways to rearrange the letters of rearrange.

The most famous one of these is the word MISSISSIPPI.
Since there are 11 total letters, with 4 "I"s, 4 "S"s, and 2 "P"s, you would work out

__11!___
4! 4! 2!

and this equals 34,560 ways to get distinguishable permutations.

I hope this helps.

Answer 3

Take items: A B C

Permute them:
ABC
ACB
BAC
BCA
CAB
CBA

These are all distinguishable from each other since they contain letters in different order.

Take items: A A C
AAC
ACA
AAC
ACA
CAA
CAA

Now, I have permuted these letters same as above (I have considered the second A as a B). However, there are 2 events: AAC, two events: ACA and two events: CAA. These 2 events are indistinguishable from each other since the orders are same.