Key Concepts and Formulas

• Permutations with Repetition: The number of distinct permutations of $n$ objects, where $p_1$ are of one kind, $p_2$ of a second kind, ..., and $p_k$ of a $k$-th kind, is given by $\frac{n!}{p_1! p_2! \dots p_k!}$.
• Bundling Method: When certain objects must always occur together, treat them as a single unit (a "block") for arrangement purposes. Remember to also account for the arrangements within the block.
• Multiplication Principle: If there are $m$ ways to do one thing and $n$ ways to do another, then there are $m \times n$ ways to do both.

Step-by-Step Solution

Step 1: Analyze the word and identify repetitions.

We are given the word "INDEPENDENCE". We need to count the occurrences of each letter to identify repetitions. This is crucial for applying the permutation formula correctly.

• I: 1
• N: 3
• D: 2
• E: 4
• P: 1
• C: 1

Total number of letters = 12.

Step 2: Separate vowels and consonants.

Next, we separate the vowels and consonants in the word. This allows us to apply the bundling method to the vowels later.

Vowels: I, E, E, E, E (5 vowels) Consonants: N, D, P, N, D, N, C (7 consonants)

Step 3: Apply the bundling method to the vowels.

Since all vowels must occur together, we treat them as a single block, 'V'. We need to find the arrangements within this block and the arrangements of the block with the consonants.

• Part A: Arrangements within the vowel block 'V'

The vowel block contains the letters (I, E, E, E, E). We need to find the number of distinct arrangements of these 5 letters. Since 'E' is repeated 4 times, we use the permutation formula for repeated items.

$\text{Arrangements within V} = \frac{5!}{4! \times 1!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{(4 \times 3 \times 2 \times 1) \times 1} = 5$

Explanation: There are 5 letters in total, with 'E' repeating 4 times and 'I' appearing once.

• Part B: Arrangements of the consonants and the vowel block 'V'

Now, we treat the vowel block 'V' as a single unit. We need to arrange the 7 consonants along with this block.

The units to arrange are: N, D, P, N, D, N, C, (V) Total units = 8

Let's list the repetitions among these 8 units:

• N: 3 times
• D: 2 times
• P: 1 time
• C: 1 time
• V: 1 time

Using the permutation formula for repeated items:

$\text{Arrangements of (Consonants + V)} = \frac{8!}{3! \times 2! \times 1! \times 1! \times 1!} = \frac{8!}{3! \times 2!} = \frac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(3 \times 2 \times 1) \times (2 \times 1)} = \frac{40320}{12} = 3360$

Explanation: We are arranging 8 units in total, where 'N' is repeated 3 times and 'D' is repeated 2 times.

Step 4: Calculate the total number of arrangements.

To find the total number of arrangements where all vowels occur together, we multiply the number of arrangements within the vowel block by the number of arrangements of the consonants and the vowel block. This follows from the multiplication principle because the arrangement of vowels within the block is independent of the arrangement of the block with the consonants.

Total arrangements = (Arrangements within V) $\times$ (Arrangements of Consonants + V) Total arrangements = $5 \times 3360 = 16800$

Common Mistakes & Tips:

• Double-check letter counts: Miscounting the frequency of letters (especially E, N, and D in this case) is a common source of error.
• Internal Arrangement: Don't forget to account for the arrangements within the bundled group of vowels.
• Remember the Formula: Ensure you use the correct permutation formula for cases with repetitions, dividing by the factorials of the repetition counts.

Summary:

We solved the problem by first counting the occurrences of each letter in "INDEPENDENCE". Then, we applied the bundling method to the vowels, treating them as a single unit. We calculated the arrangements within the vowel block and the arrangements of the consonants along with the vowel block. Finally, we multiplied these two results to get the total number of arrangements satisfying the given condition.

The final answer is $\boxed{16800}$, which corresponds to option (A).