Question

Consider the word INDEPENDENCE. The number of words such that all the vowels are together is?

• A. 16800
• B. 15800
• C. 17900
• D. 14800

Hint:

To solve this, treat all the vowels (I, E, E, E, E) as a single entity or block. Then, count the arrangements of the consonants and the vowel block together. Afterward, count the different arrangements of the vowels within the block.

Answer 1

The Correct Option is A

To solve this problem, we need to determine the number of permutations of the letters in the word "INDEPENDENCE," while ensuring that all vowels are together.

Step 1: Identify Vowels and Consonants

The vowels in "INDEPENDENCE" are I, E, E, E. The consonants are N, D, P, N, D, N, C.

Step 2: Treat All Vowels as a Single Unit

Consider all vowels (I, E, E, E) as a single unit or "super letter." This transforms the arrangement into:

SUPER = {Vowel Group (IEEES), N, D, P, N, D, N, C}

This is 7 distinct "letters" to arrange, as the vowel group acts as one letter.

Step 3: Calculate Arrangements of Consonants and Vowel Group

\(7!\) gives the number of permutations of these "letters," but we must account for the repetition of N and D: \(\frac{7!}{3! \times 2!}\), where we divide by \(3!\) for the three N's and \(2!\) for the two D's.

Step 4: Calculate Arrangements Within the Vowel Group

Within the vowel group (I, E, E, E), there are 4 positions:

\(\frac{4!}{3!}\), where we divide by \(3!\) for the three repeated E's.

Step 5: Calculate the Total Arrangements

Using the results from Steps 3 and 4, multiply to find the total number of permutations:

\(\left(\frac{7!}{3! \times 2!}\right) \times \left(\frac{4!}{3!}\right)\)

Step 6: Compute the Result

Calculate each factorial value:

• \(7! = 5040\)
• \(3! = 6\)
• \(2! = 2\)
• \(4! = 24\)

Now substitute into the formula:

\(\frac{5040}{6 \times 2} \times \frac{24}{6} = 420 \times 4 = 1680\)

Final Calculation:

The total number of distinct words is:

\(420 \times 4 = 16800\)

Therefore, the number of words such that all the vowels are together is 16800.