Question

How many words can be formed with the letters of the word DAUGHTER,vowel remaining always together?

Answer 1

For the letters of words DAUGHTER .

We separate the vowels and consonants, i.e., D G H T R And A U E

Lets consider the vowels A U E as a single letter V.

Note that Now we have grouped all the vowels together…

So the Letters to be arranged are D G H T R V

Now there are 6! ways of arranging the letters DGHTRV .

Now, We know that V consists of 3 letters A U E. These three letters can interchange within themselves. So no. of permutations of V is 3!.

Now total no. of arrangements of the Letters of word daughter so that all vowels always come together is, by fundamental principle of counting = 6! x 3!.

Thus, the total number of words formed will be equal to (6!∗3!)=4320

Answer 2

Hello friends!!

In given word, we treat the vowels AUE as one letter.

Thus, we have DGHTR(AUE), 6 letters.  

                                OR

The vowels are ‘a, u, e’ and the consonants are “d, g, h, t, r”.

Now, all the vowels should come together, so consider the bundle of vowels as one letter, then total letters will be 6.

So, no.  of ways of arranging these letters = 6! = 720.

Now,  3 vowels can be arranged themselves in 3! = 6.

Required number of ways = 720*6 = 4320.  

I hope this will help you.

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