Question

The number of words (with or without meaning) that can be formed from all the letters of the word “LETTER” in which vowels never come together is

Answer 1

Find all the possible scenarios where vowels come together and subtract it from the total words possible

The word LETTER has $6$ alphabets of which $2$ are repeated twice.

The total number of $6$ letter words $=\frac{6!}{2!2!}=\frac{6\times 5\times 4\times 3\times 2}{2\times 2}$

$=180$ words

There are $2$ vowels in the word LETTER, that is, (EE).

If EE are assumed to be together they can be arranged in $\frac{5!}{2!}$ ways.

So, the words where vowels are together are $\frac{5!}{2!}=\frac{120}{2}=60$

Thus, the total number of words where vowels are not together$=180-60=120$

Hence, there are $120$ words formed from the word LETTER where no vowels are together.