Questions: Consider the word AARDVARK. Determine all the permutations of the letters in which the three A's are together.

Solution Steps

To find the permutations of the letters in "AARDVARK" where the three A's are together, we treat the three A's as a single block, denoted as \( AAA \). This reduces the problem to permuting the letters \( \{ AAA, R, D, V, R, K \} \).

The new set of letters consists of 6 items: \( AAA, R, D, V, R, K \). The letter counts are as follows:

• \( AAA: 1 \)
• \( R: 2 \)
• \( D: 1 \)
• \( V: 1 \)
• \( K: 1 \)

The total number of permutations of these letters can be calculated using the formula for permutations of multiset:

\[ \text{Total Permutations} = \frac{n!}{n_1! \cdot n_2! \cdot n_3! \cdots} \]

Where \( n \) is the total number of items, and \( n_1, n_2, \ldots \) are the counts of each distinct item. Here, we have:

\[ n = 6, \quad n_R = 2, \quad n_{AAA} = 1, \quad n_D = 1, \quad n_V = 1, \quad n_K = 1 \]

Thus, the calculation becomes:

\[ \text{Total Permutations} = \frac{6!}{2! \cdot 1! \cdot 1! \cdot 1! \cdot 1!} = \frac{720}{2} = 360 \]

Final Answer