Question

How many words can be formed from the letters of the word 'DAUGHTER' so that
(i) the vowels always come together? (ii) the vowels never come together?

Answer 1

The letters of the word daughter are $“d,a,u,g,h,t,e,r”.$

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

$\left(i\right)$Now, all the vowels should come together, so consider the bundle of vowels as one letter, then total letters will be $6$.

So, the number of words formed by these letters will be $6!$

but, the vowels can be arranged differently in the bundle, resulting in different words, so we have to consider the arrangements of the $3$ vowels.

So, the arrangements of vowels will be $3!$

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

$\left(ii\right)$First arrange $5$ consonants in five places in $5!$ ways.

$6$ gaps are created. Out of these $6$ gaps, select $3$ gaps in${}^6{C}_3$ ways and then make the vowels permute in those $3$ selected places in $3!$ ways.

This leads $5!{\times }^6{C}_3\times 3!=14400.$