I see similar questions asked on here and obviously I did some research and read my book, but it seems like every explanation contradicts another in some way. There are basically infinite scenarios using these and every example problem/scenario I seem to convince myself it could be both!
Here are some of my understandings of each:
Permutation: Every detail matters and ALL ways of doing something. "Think of permutations as a list."
Combinations: Used for groups. Order and Position DOES NOT matter.
My Confusion:
a.) If permutations are ALL ways of doing something.. then why does order/position/type matter?
b.) If order does NOT matter with combinations.. why are "Locks" said to have a "combination" when clearly the order does matter with a lock? If the "combination" to unlock something is 1-2-3.. then clearly 1-3-2 would not work. Therefore it seems like order does matter..
c.) If permutations are ALL ways of doing something and if EVERY detail matters.. then why are the number of permutations larger than the number of combinations?
Sorry if I included too much. I'm really struggling with this and every time I think I understand a scenario/problem.. I look at another and have no idea how to do it! I'd greatly appreciate any help. Thank you!
Best Answer
a) this one depends on the question at hand.
Say you have 3 people (A, B, C) and you want to put them in a row. Finding all the arrangements would include
A B C
A C B
B C A
B A C
C A B
C B A
3P3 = 3! = 6 arrangements
but the same three people were in there so if order didn't matter, this would only count as 1 arrangement since in each case there was 1A, 1B, 1C.
that's how a combination is different
$3\choose3$ = 1 arrangement
b) You're right that's clever. Locks don't have permutations or combinations actually
you can't say a lock has 10P3 = 10 x 9 x 8
A lock has different numbers for each position
so n(S) = 10x10x10
c) The combination is just the number of choices provided we are not ordering the items but choosing a certain amount of them and grouping them together. A permutation is all the ways of arranging all the combinations into specific orders like in a)
A question on combinations.
If there are 30 people in a class and you need to pick 2 people to clean up at the end of the day. How many ways can you choose those people. This is one where order doesn't matter because picking Eddie and Fred is the same as Fred and Eddie
n(S) = $30\choose2$ = 435 (not 30P2 = 870. Note that 870 is double 435 so see how in this case you would have had 870 full arrangements but only half of them are valid choices. Why is that? Obviously because for every Fred and Eddie there was a Fred Eddie)
If you are choosing 3 people it gets worse because
30P3 = 24 360 but $30\choose3$ is 4 060. Clearly you cannot just divide the permutation by 3 to get 4060. You need to divide it by 3! because there will be
Fred, Eddie, Freddie.
Fred, Freddie, Eddie.
Eddie, Fred, Freddie.
Eddie, Freddie, Fred.
Freddie, Fred, Eddie.
Freddie, Eddie, Fred.
$A\chooseB$ = $A\PermuteB$ / B!