Question

The number of ways in which the letters of the word "INSURANCE" be arranged so that the vowels are never separated is:

• A. $4320$
• B. $8640$
• C. $21600$
• D. $10300$

Answer 1

The correct option is B $8640$

Vowels are $IUAE$
Total letters $=9$
Repetitions $=2$ $($ $N$'s$)$
Consider the band of vowels as one.
Then technically we are left with $(9-4)=5$ letters
Hence, arrangement of $6$ letters in which one letter is repeated twice is $($$5$ letters and $1$ band of vowels$)$$=\frac{6!}{2!}$$=360$
Now, the vowels will arrange within themselves in $4!$ ways.
Hence, the required permutation will be $=360.4!$
$=360\left(24\right)$
$=8640$