In how many arrangements of the word ‘GOLDEN’ will the vowels never occur together?
To find: number of words
Condition: vowels should never occur together.
There are 6 letters in the word GOLDEN in which there are 2 vowels.
Total number of words in which vowels never come together =
Total number of words - total number of words in which the vowels come together.
A total number of words is $6 !=720$ words.
Consider the vowels as a group.
Hence there are 5 groups that can be arranged in $P(5,5)$ ways, and vowels can be arranged in $\mathrm{P}(2,2,$, ways.
Formula:
Number of permutations of $n$ distinct objects among $r$ different places, where repetition is not allowed, is
$P(n, r)=n ! /(n-r) !$
Total arrangements $=P(5,5) \times P(2,2)=\frac{5 !}{(5-5) !} \times \frac{2 !}{(2-2) !}$
$=\frac{5 !}{0 !} \times \frac{2 !}{0 !}=120 \times 2=240$
Hence a total number of words having vowels together is 240.
Therefore, the number of words in which vowels don't come together is $720-240=480$ words.