Count the letters in ASSASSINATION

The word is: $\text{A S S A S S I N A T I O N}$

Total letters $= 13$.

Their frequencies are:

• $A = 3$
• $S = 4$
• $I = 2$
• $N = 2$
• $T = 1$
• $O = 1$

Identify the vowels

Vowels are: $A, A, A, I, I, O$ So total vowels $= 6$.

Consonants are: $S, S, S, S, N, N, T$ So total consonants $= 7$.

Use the condition that all vowels occur together

Treat all the vowels as one single block.

Then the objects to arrange are:

• 1 vowel block
• $S, S, S, S$
• $N, N$
• $T$

So total objects $= 8$.

These 8 objects have repetitions among consonants:

• $S$ repeated $4$ times
• $N$ repeated $2$ times

Therefore, number of arrangements of these 8 objects is $\frac{8!}{4!\,2!}$

Arrange the vowels within the vowel block

The vowels are: $A, A, A, I, I, O$

Number of distinct arrangements of these 6 vowels is $\frac{6!}{3!\,2!}$

Multiply both counts

Total number of required words is $\frac{8!}{4!\,2!}\times \frac{6!}{3!\,2!}$

Now compute: $\frac{8!}{4!\,2!} = \frac{40320}{24\cdot 2} = \frac{40320}{48} = 840$

and $\frac{6!}{3!\,2!} = \frac{720}{6\cdot 2} = \frac{720}{12} = 60$

Hence, $840 \times 60 = 50400$

Final answer

$\boxed{50400}$

Comparison with stored correct answer

Stored correct answer = $50400$.

This matches the derived answer.