Key Concepts and Formulas

• Permutations with Repetition: The number of permutations of $n$ objects, where there are $n_1$ identical objects of type 1, $n_2$ identical objects of type 2, ..., and $n_k$ identical objects of type k, is given by $\frac{n!}{n_1!n_2!...n_k!}$.
• Block Method: When a group of items must always be together, treat them as a single block. Calculate the arrangements within the block and then arrange the block with the remaining items.
• Multiplication Principle: If one event can occur in $m$ ways and another independent event can occur in $n$ ways, then the two events can occur in $m \times n$ ways.

Step-by-Step Solution

Step 1: Analyze the Letters of the Word ASSASSINATION

We need to determine the frequency of each letter in the word ASSASSINATION and identify the vowels and consonants. This is crucial for applying the block method and accounting for repetitions.

The word ASSASSINATION has 13 letters:

• Vowels: A (3 times), I (2 times), O (1 time). Total vowels: 6
• Consonants: S (4 times), N (2 times), T (1 time). Total consonants: 7

Step 2: Form the Vowel Block

Since the vowels must occur together, we treat the 6 vowels (A, A, A, I, I, O) as a single block. Let's denote this block as 'V'.

Explanation: This simplifies the problem by reducing the number of individual items we need to arrange.

Step 3: Calculate the Number of Internal Arrangements within the Vowel Block

We need to find the number of ways to arrange the vowels A, A, A, I, I, and O within the vowel block. This involves permutations with repetition.

We have 6 vowels in total, with A repeating 3 times and I repeating 2 times. Using the formula for permutations with repetition:

$\text{Internal arrangements} = \frac{6!}{3!2!1!}$

Calculating the factorials:

• $6! = 720$
• $3! = 6$
• $2! = 2$
• $1! = 1$

Substituting the values: $\text{Internal arrangements} = \frac{720}{6 \times 2 \times 1} = \frac{720}{12} = 60$

Explanation: There are 60 different ways to arrange the vowels within the vowel block.

Step 4: Arrange the Vowel Block and the Consonants

Now we need to arrange the vowel block (V) and the 7 consonants (S, S, S, S, N, N, T). This gives us a total of 8 items to arrange.

We have the following items: V, S, S, S, S, N, N, T. Applying the permutations with repetition formula:

$\text{Arrangements of block and consonants} = \frac{8!}{4!2!1!1!}$

Calculating the factorials:

• $8! = 40320$
• $4! = 24$
• $2! = 2$
• $1! = 1$

Substituting the values: $\text{Arrangements of block and consonants} = \frac{40320}{24 \times 2 \times 1 \times 1} = \frac{40320}{48} = 840$

Explanation: There are 840 ways to arrange the vowel block and the consonants.

Step 5: Combine the Results using the Multiplication Principle

To find the total number of words, we multiply the number of internal arrangements of the vowel block by the number of arrangements of the vowel block and consonants:

$\text{Total words} = (\text{Internal arrangements}) \times (\text{Arrangements of block and consonants})$ $\text{Total words} = 60 \times 840 = 50400$

Explanation: For each arrangement of the vowel block and consonants, there are 60 possible arrangements of the vowels within the block.

Step 6: Find the Number of Trailing Zeroes

The question asks for the number of trailing zeroes in the calculated value. 50400 = 504 * 100. Therefore, there are 2 trailing zeroes.

Step 7: Extract the number before the Trailing Zeroes 50400 = 504 * 100. The number before the trailing zeroes is 504.

Step 8: Apply the Logarithm Base 100 on the result of the previous step log base 100 of 504. We need to find the greatest integer less than or equal to this logarithm. Since $100^2 = 10000$, $log_{100}(504)$ is between 1 and 2. Since $100^1 = 100$, $log_{100}(504) > 1$. The greatest integer less than or equal to $log_{100}(504)$ is 1.

Step 9: Add 7 to the result of the previous step 1 + 7 = 8.

Common Mistakes & Tips

• Forgetting Internal Arrangements: Always remember to calculate and include the arrangements within the vowel block.
• Miscounting Repetitions: Ensure accurate counting of repeated letters in both internal and external arrangements.
• Incorrect Application of Multiplication Principle: Understand why the counts are multiplied. The events must be independent.

Summary

By treating the vowels as a single block and accounting for internal arrangements and repetitions, we found the total number of words that can be formed from the letters of ASSASSINATION with the vowels together is 50400. Applying logarithm base 100 to 504, adding 7, and finding the greatest integer, we arrive at 8.

The final answer is $\boxed{8}$.