Calculating Arrangements of INDEPENDENCE with Vowels Together

The problem asks for the number of distinct arrangements of the letters in the word INDEPENDENCE such that all the vowels appear together.

Analyzing the Letters in INDEPENDENCE

The word INDEPENDENCE has 12 letters. Let's count the frequency of each letter:

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

Total letters = \(1 + 3 + 2 + 4 + 1 + 1 = 12\).

Identifying Vowels and Consonants

The vowels in the word are I, E, E, E, E. There are 5 vowels in total.

The consonants are N, N, N, D, D, P, C. There are 7 consonants in total.

Approach: Treating Vowels as a Single Unit

To ensure all vowels always occur together, we can treat the entire group of vowels (I, E, E, E, E) as a single block or unit. Now, we are arranging this single vowel block along with the individual consonants.

The units we need to arrange are:

• The vowel block: (I E E E E)
• The consonants: N, N, N, D, D, P, C

This gives us a total of \(1 + 7 = 8\) units to arrange.

Step-by-Step Calculation for Arrangements

Step 1: Arrange the 8 Units (Vowel Block + Consonants)

We have 8 units to arrange. These units include repetitions:

• The vowel block (1 unit)
• N (3 times)
• D (2 times)
• P (1 time)
• C (1 time)

The number of ways to arrange these 8 units is given by the permutation formula for objects with repetitions:

\[ \text{Number of arrangements} = \frac{\text{(Total number of units)}!}{\text{(Count of N)!} \times \text{(Count of D)!} \times \text{(Count of P)!} \times \text{(Count of C)!} \times \text{(Count of Vowel Block)!}} \]

\[ = \frac{8!}{3! \times 2! \times 1! \times 1! \times 1!} = \frac{8!}{3! \times 2!} \]

Let's calculate this value:

\[ \frac{8!}{3! \times 2!} = \frac{40320}{(6) \times (2)} = \frac{40320}{12} = 3360 \]

There are 3360 ways to arrange the 8 units.

Step 2: Arrange the Letters Within the Vowel Block

The vowel block consists of the letters I, E, E, E, E. There are 5 letters in this block, with the letter E repeated 4 times.

The number of ways to arrange these 5 letters within the block is:

\[ \text{Number of arrangements within vowel block} = \frac{\text{(Total number of vowels)}!}{\text{(Count of E)!} \times \text{(Count of I)!}} \]

\[ = \frac{5!}{4! \times 1!} = \frac{5!}{4!} \]

Let's calculate this value:

\[ \frac{5!}{4!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{4 \times 3 \times 2 \times 1} = 5 \]

There are 5 ways to arrange the letters within the vowel block.

Step 3: Combine the Arrangements

To find the total number of arrangements where all vowels are together, we multiply the number of ways to arrange the 8 units by the number of ways to arrange the letters within the vowel block.

\[ \text{Total arrangements} = (\text{Arrangements of units}) \times (\text{Arrangements within vowel block}) \]

\[ = 3360 \times 5 \]

\[ = 16800 \]

Thus, there are 16,800 arrangements of the letters of the word INDEPENDENCE where all the vowels always occur together.

Conclusion on INDEPENDENCE Arrangements

The quantity of arrangements of the letters of the word INDEPENDENCE where all the vowels always occur together is 16,800.

Additional Information: Counting Principles in Combinatorics

This problem utilizes fundamental counting principles from combinatorics, specifically permutations with repetition. Understanding how to handle identical items is crucial when calculating the number of distinct arrangements.

• Permutation: An arrangement of objects in a specific order.
• Combination: A selection of objects where the order does not matter.
• Permutations with Repetition: When dealing with a set of objects where some are identical, the total number of distinct permutations is reduced. The formula accounts for the indistinguishable items by dividing the total permutations (if all were distinct) by the factorial of the counts of each repeated item.

Grouping items together, as done with the vowels in this problem, is a common technique to handle constraints in permutation and combination problems. The grouped items are treated as a single unit, and then the arrangements within that unit are considered separately.