The number of words, with or without meaning, that can be formed...

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

Now, we have \left(AAAIIO\right)SSSSNNT$</p><p>This has$8 = ( 7 + 1 )$letters of which$S$occurs$4$times, and$N$occur$2$times,</p><p>Thus, the number of ways of arranging these letters$= \dfrac{8 !}{4 ! 2 !} = 840$.</p><p>Now,$6$vowels in which A$ occurs $3$times, and $Ioccur two times, they can be arranged in$\dfrac{6 !}{3 ! 2 !} =$60$\text{ways.}

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