Mathematics - Permutation & Combination Question with Solution | TestHub

In the word $\mathrm{ASSASSINATION}$, we will treat all the vowels that is $\mathrm{AAAIIO}$  as one letter only.

Now, we have $\left(\mathrm{AAAIIO}\right)\mathrm{SSSSNNT}$

This has $8=(7 + 1)$ letters of which $\mathrm{S}$ occurs $4$ times, and $\mathrm{N}$ occur $2$ times,

Thus, the number of ways of  arranging these letters $=\frac{8!}{4!2!}=840$.

Now, $6$ vowels in which $\mathrm{A}$ occurs $3$times, and $\mathrm{I}$ occur two times, they can be arranged in $\frac{6!}{3!2!}=60 \text{ ways. }$

 Therefore, required number of ways$= (840 \times 60) = 50400.$