Question

In how many ways can the letters of the word $PERMUTATIONS$ be arranged if the vowels are together.

Answer 1

There are $12$ letters in the word $PERMUTATIONS$ which have $T$ two times.

Now, the vowels $a,e,i,o,u$ are together. Let it be considered in one block.

The letters of vowels can be arranged in $5!$ ways.

Thus there are $7$ letters and $1$ block of vowels with $T$ two times.

Therefore,

Number of arrangements = $\frac{8!}{2!}\times 5!=2419200$