[RESOLVED] Permutations & Combinations

[Yunie]

(a) Find the number of arrangements of the letters in the word ANGLES if (i) there is no restriction.& (ii) the vowels must be separated.

Please show your workings and explain. Thanks a lot.

[krtxmrtz]

Without restriction it's 6! = 720

If the 2 vowels must be separated, then it's 720 minus the number of cases where the vowels are NOT separated. The latter are very few and can be sketched out as:

AE---
EA---
-AE--
-EA--
--AE-
--EA-
---EA
---EA

where - represents any other letter.

So, its 720 - 8 = 712

[Logophobic]

There are many more than 8 permutations where the vowels are not separated:
AENGLS
AENGSL
AENLGS
etc...

Taking the vowels together as a single 'character', we find 5! = 120 permutations without regarding the order of the vowels. Since there are 2 permutations of the vowels, the total number of permutations where the vowels are not separated is 240. Thus, the vowels are separated in 480 of the 720 permutations.

[Yunie]

Thanks guys! I understood the question!

[Yunie]

Wait! SO that means, everytime when we are doing the above type of questions or anything that has to do with permutations & combinations question, we have to think of grouping like-terms together (eg. in this case, we group the 'vowels' together) right?

Thanks again.

[krtxmrtz]

There are many more than 8 permutations where the vowels are not separated:
AENGLS
AENGSL
AENLGS
etc...

Taking the vowels together as a single 'character', we find 5! = 120 permutations without regarding the order of the vowels. Since there are 2 permutations of the vowels, the total number of permutations where the vowels are not separated is 240. Thus, the vowels are separated in 480 of the 720 permutations.

Oh my, I forgot about the permutations of the other 3 characters when the 2 vowels are together... Good for you!

Yunie, sorry if I misled you.

[krtxmrtz]

Wait! SO that means, everytime when we are doing the above type of questions or anything that has to do with permutations & combinations question, we have to think of grouping like-terms together (eg. in this case, we group the 'vowels' together) right?

Thanks again.

Aside from the fact that I had overlooked those other cases, Logophobic's logic is more like what you should do. This type of problems should not be solved by writing down all possible cases -usually there are so many you can't anyway. But sometimes it helps understand the problem.

[Yunie]

I got it, thanks krtxmrtz!