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Question: Find the number of arrangements of the letters of the word $INDEPENDENCE$. In how many of these arra...

Find the number of arrangements of the letters of the word INDEPENDENCEINDEPENDENCE. In how many of these arrangements, do the vowels never occur together

A

1646400

B

1846800

C

1546400

D

1606500

Answer

1646400

Explanation

Solution

To find the number of arrangements where the vowels never occur together, we first identify the letters and their frequencies in the word "INDEPENDENCE".

The word "INDEPENDENCE" has 12 letters:

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

The vowels are I, E, E, E, E (5 vowels). The consonants are N, N, N, D, D, P, C (7 consonants).

Step 1: Calculate the total number of distinct arrangements of the letters. The total number of letters is 12. The letter 'N' repeats 3 times. The letter 'D' repeats 2 times. The letter 'E' repeats 4 times.

Total arrangements = 12!3!×2!×4!=479001600(6)×(2)×(24)=479001600288=1663200\frac{12!}{3! \times 2! \times 4!} = \frac{479001600}{(6) \times (2) \times (24)} = \frac{479001600}{288} = 1663200.

Step 2: Calculate the number of arrangements where all vowels occur together. Treat all 5 vowels (I, E, E, E, E) as a single block. First, find the number of ways to arrange the vowels within this block: Number of arrangements of (I E E E E) = 5!4!×1!=12024=5\frac{5!}{4! \times 1!} = \frac{120}{24} = 5.

Now, consider this block of vowels as a single unit. We are arranging this vowel block along with the 7 consonants: N, N, N, D, D, P, C. So, we have 1 (vowel block) + 7 (consonants) = 8 units to arrange. The repeated units among these 8 are 'N' (3 times) and 'D' (2 times).

Number of arrangements of these 8 units = 8!3!×2!=403206×2=4032012=3360\frac{8!}{3! \times 2!} = \frac{40320}{6 \times 2} = \frac{40320}{12} = 3360.

The total number of arrangements where all vowels occur together is the product of the arrangements of the units and the arrangements within the vowel block: Arrangements (vowels together) = 3360×5=168003360 \times 5 = 16800.

Step 3: Calculate the number of arrangements where the vowels never occur together.

Number of arrangements (vowels never occur together) = (Total arrangements) - (Arrangements where all vowels occur together) =1663200−16800=1646400= 1663200 - 16800 = 1646400.

This matches option (A).

The final answer is 1646400\boxed{\text{1646400}}.