(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. :)
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(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. :)
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
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.
Thanks guys! I understood the question! :)
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. :)
Oh my, I forgot about the permutations of the other 3 characters when the 2 vowels are together... :o Good for you!Quote:
Originally Posted by Logophobic
Yunie, sorry if I misled you.
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.Quote:
Originally Posted by Yunie
I got it, thanks krtxmrtz! :)