Question

In how many ways can the letter of the word $PERMUTATIONS$ can be arranged so that all the vowels come together

Answer 1

There are total $12$ letters in the word $PERMUTATIONS$, with $T$ repeated twice.
Number of vowels in the given word are $5$.
Since vowels have to always occur together, so they are considered as a single object . This single object (letter) together with the remaining $7$ letters will give us $8$ objects (letters).
These $8$ objects in which there are $2$ $T^{\prime }s$ can be arranged in $\frac{8!}{2!}$ ways .Corresponding to each of these arrangements, the $5$ different vowels can be arranged in $5!$ ways.
Therefore, by multiplication principle, required number of arrangements in this case $=\frac{8!}{2!}\times 5!=2419200$