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Permutation and Combination

Ruth has the following set of refrigerator magnets: {A,B,C,D,E,F,G}.

a. How many different three-letter strings can she form with these magnets ?
Ans. P(7,3)=210

b. How many different three-letter strings can she form if the middle letter must be a vowel ?
Ans. 2.6.5=60

My question is why the answer of part A is P. There is nothing which it says about order ?

I dont get the part B question

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2 Solutions
you're right that the permutation is related to ordering of the elements.

The ordering is implied by the phrase "three-letter strings "

the string "ABC" is different than the string "BCA"  because a character string is defined by the order of the characters within it.

if the question had set "sets of 3 letters"  then ordering wouldn't be a factor and you'd use combinations instead.

I think the 2.6.5  is supposed to mean  "."  as multiplication  so  2 x 6 x  5 = 60

the reason B is 60 is because

you have 2 choices for the middle position: A or E  

after choosing one of those you have 6 characters left for the first position
after choosing one of those you have 5 characters left for the last position

so 2 choices x 6 choices x 5 choices = 60 possible strings
part a 'There is nothing which it says about order ?" that is correct

This sounds like home work, so I annot give you the answer. But haow are you attacking it.
How many volws are there?

How many letters are before (and after the middle vowel
How many possibilities are there for the first half of the word.? Do not look for an equation, just count them.

How many possibilities are there for the last half of the word (remember that one letter has already been used.

Now that you have the first and last part of the word, note that you have a set with "A" in the middle and a set with "E" in the middle.
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mustish1Author Commented:
its not the homework i already have all the answers. trying to understand why its a P not C
the reason is as I stated above.  The ordering criteria isn't specified explicitly, it's simply implied.

a string in math and computer jargon is a sequence of characters.  

a sequence is an ordered list.

so a three-letter string  is an ordered list of 3 letters.

this question is fairly clear as to how it is to be interpreted, first by word choice and then confirmed by the answers provided.

if your "real" question is how do you determine permutation vs combination for any phrasing - well, there is no guarantee you'll get it right every time.  human language grammar and math don't always synch nicely.
however, in academic settings you likely wiill be allowed to ask for confirmation of intended ordering if the wording is ambiguous.
in real-life,  you can often calculate both and only one will make sense.
or simply compare some permutation/combinations and see which interpretation fits best

for example take this question.

if you arrange the 7 letters above into "words" on your refigerator, would you say  "BAG" and "GAB" are the same word?  No, probably not, so ordering matters, therefore it's a permutation.


Select three letters (say ABC). You could have selected CBA  CBA contains the same letters as ABC but in a different order Therefor CBA is a different Permutation of the letters ABC but it is the same combination of the selected set of letters.
ABC BCA CAB are all the same set of letters but the different order makes them different permutations of the same set.

(note: sdstuber: answered my questions while I was writing them so there is your part b

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