In how many different ways can the letters of the word ‘TRANSPIRATION’ be arranged so that the vowels always come together?

Solution(By Examveda Team)

The word ‘TRANSPIRATION’ has 13 letters in which each of T, R, A, N and I has come two times
We have to arrange TT RR NN PS (AA II O)
There are five vowels in the given words.
∴ We consider these give vowels as one letter.
∴ Required number of arrangements
$$\eqalign{ & = \frac{{9! \times 5!}}{{2!\, 2! \,2! \,2! \,2!}} \cr & = \frac{{9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 5 \times 4 \times 3 \times 2}}{{2 \times 2 \times 2 \times 2 \times 2}} \cr & = 1360800 \cr} $$