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In each of above possible ways (10 in total) of placing vowels, you can switch vowels around 3! ways: 3! = 6
Number of ways of arranging consonants: 4! = 24
Since there are 10 ways of arranging vowels: 10(3!*4!) = 1440
Sorry I had to edit my answer because I forgot about consonants
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Wow, quick on correcting mistakes. There was no answer when I started typing and 4 of them popped up by the time I finished! You guys are quick!
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Total ways to arrange = 7!
there are 3 vowels
ways to pick 2 vowels out of 3 is 3C2 = 3
Ways to arrange the picked two vowels = 2
Ways to arrange 2 vowel combination with other 5 letters = 6!
Total ways to arrange 7 letters with 2 vowels togather = 6 * 6!

Total ways to arrange 7 letter with vowels seperated = 6! = 720

No of ways in which vowels are together = no of ways in which exactly two vowels are together + no of ways in which 3 vowels are together.

But if you just choose 2 vowels in 3C2 and consider it as a set , you will have this set, one remaining vowel and 4 other consonants.
If you rearrange these it also covers set containgin 3 vowels together.
so you have 2 * 3C2 * 5! as invalid combinations

No of ways in which vowels are together = no of ways in which exactly two vowels are together + no of ways in which 3 vowels are together.

But if you just choose 2 vowels in 3C2 and consider it as a set , you will have this set, one remaining vowel and 4 other consonants. If you rearrange these it also covers set containgin 3 vowels together. so you have 2 * 3C2 * 5! as invalid combinations

No of ways to arrange 7 letters = 7! = 5040

Desired ways = 5040 - 2 * 3C2 * 5! = 4320

Anand,

Just wondering if ur solution takes into account the different ways in which 3 vowels will be together and there placement along with respect to consonants.

Total ways to arrange = 7! there are 3 vowels ways to pick 2 vowels out of 3 is 3C2 = 3 Ways to arrange the picked two vowels = 2 Ways to arrange 2 vowel combination with other 5 letters = 6! Total ways to arrange 7 letters with 2 vowels togather = 6 * 6!

Total ways to arrange 7 letter with vowels seperated = 6! = 720

well, you assumed words such as xxVyVza have no vowles together. For example, x, y or z can be the one Vowel you didn't choose.

No of ways in which vowels are together = no of ways in which exactly two vowels are together + no of ways in which 3 vowels are together.

But if you just choose 2 vowels in 3C2 and consider it as a set , you will have this set, one remaining vowel and 4 other consonants. If you rearrange these it also covers set containgin 3 vowels together. so you have 2 * 3C2 * 5! as invalid combinations

No of ways to arrange 7 letters = 7! = 5040

Desired ways = 5040 - 2 * 3C2 * 5! = 4320

Your method makes the most sense. Yes 4320 should be the correct answer. Thanks

word PROMISE has vowels O,I,E
if you chose two vowels then you will have following combinations

OI - it can also also be arranged as IO
IE - it can also be arranged as EI
OE - it can also be arranged as EO

Consider one combination OI
Now you have PRMSE and OI
Following are some of the combinations that can be obtained
ESMRPOI
ESMRPIO
and
SMRPEIO - > Here all the 3 vowels are together
So just accounting for two vowels at a time should cover everything.

Thus for each vowel pair we have invalid combinations as
2 * 5! -> for OI and IO
2 * 5! -> for IE and EI
2 * 5! -> for EO and OE
this is same as 2 * 3C2 * 5! = 720

Total combinations are 7! = 5040
so valid combinations are 7! - 720 = 4320