How many different ways can the letter of Word Champion be arranged so that all the vowels come together?

= 120 ways. The vowels (EAI) can be arranged among themselves in 3!

How many arrangements are there where no two vowels are next to each other?

ways. In total we have (63)×3! ×5! =14400 ways.

How many words can be formed from the letters of the word rainbow with A and B on either end?

This result 4320 is overcounting since it is only considering the fact that all vowels come together only and not the case where only two vowels can also come together also, as the question says vowels are never together, and not that all the vowels are not together only. This is why the correct answer is 1440.

How many ways the word rear can be arranged in which vowels never comes together?

Answer 2 D. No. of permutations possible with vowels never together = 360-120 = 240.

How many ways can the vowels (OIA) be arranged among themselves?

When the vowels OIA are always together, they can be supposed to form one letter. Then, we have to arrange the letters PTCL (OIA). Now, 5 letters can be arranged in 5! = 120 ways. The vowels (OIA) can be arranged among themselves in 3! = 6 ways. Therefore Required number of ways = (120 x 6) = 720.

What is the total number of groups when the vowels are one?

The total number of groups when the vowels are one group and the rest are individuals= 4! First of all we have to find all vowels occur together There are 6 different letters in the word ORANGE, in which there are 3 vowels, namely, O, A and E.

How many different ways can the letters of the word ‘orange’ be arranged?

Now we need In how many different ways can the letters of the word ‘ORANGE’ be arranged so that the three vowels never come together so If we have to count those permutations in which all vowels are never together, we first have to find all possible arrangements of 6 letters taken all at a time,which can be done in 6! ways.

How many permutations of a word can be possible with 3 vowels?

No. of ways 3 vowels can occur in 4 different places = 4 P 3 = 24 ways. After 3 vowels take 3 places, no. of ways 4 consonants can take 4 places = 4 P 4 = 4! = 24 ways. Therefore, total number of permutations possible = 24*24 = 576 ways.