Question

How many words can be formed of the letters in the word LAUGHTER, so that the vowels are not separated?

Answer 1

Hint: In alphabets there are 6 vowels a,e,i,o,u . So in word LAUGHTER there are 3 vowels a,e,u.In general the number of ways to arrange n letters in different ways is n!.

Complete step-by-step answer:
Given word is ‘LAUGHTER’
Total number of letters in a given word is 8 in which 3 letters are vowels.
As we need to keep all three vowels together. So we will count all 3 vowels together as a single letter.
Now we have a total of 6 letters which we can arrange in $6!$ ways but we can also arrange 3 vowels in $3!$ ways.
Hence number of words can be formed of the letters in the word LAUGHTER, so that the vowels being not separated is
$\Rightarrow 6!\times 3!$
$\Rightarrow 6\times 5\times 4\times 3\times 2\times 1\times 3\times 2\times 1$
$\Rightarrow 4320$.

Note:In general, the factorial of n can be defined as a product of all integers from n to 1. We can write it as
$n!=n(n-1)(n-2)(n-3)(n-4)........................3.2.1$
It is defined only for positive integers.
Here we need to remember that we are arranging 3 vowels together. So we also need to consider the number of ways from which we can arrange three vowels.