If the letters of word DAUGHTER are to be arranged so that vowel occupy the odd places then number of different words are?

Question

Answer

There are 8 letters in the word daughter of which 3 are vowels and 5 are consonants. Again, there are 4 odd places viz 1st, 3rd,5th & 7th and 4 even places viz. 2nd, 4th, 6th & 8th. Now, in 4 odd places 3 vowels can be put in 4P3 = 24 ways. Having fixed up the vowels in 4 odd places, we are left with 5 places viz 1 odd and 4 even places. In these 5 places we have to fix 5 consonants which can be done in 5P5 = 5! = 120 ways. Hence, total number of arrangements (or words) will be = 24 x 120 = 2880.

Answer

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Answer

Arranging Letters of Word DAUGHTER with Vowels in Odd Places

Understanding the Problem

To solve this problem, we need to arrange the letters of the word DAUGHTER in such a way that the vowels occupy the odd places.

Identifying the Vowels

The first step is to identify the vowels in the word DAUGHTER. The vowels in this word are A, U, and E.

Determining the Number of Odd Places

Since we want the vowels to occupy the odd places, we need to determine the number of odd places. In this case, there are four odd places, which are the first, third, fifth, and seventh positions.

Using Permutations

We can use permutations to determine the number of different words that can be formed by arranging the letters of the word DAUGHTER such that the vowels occupy the odd places.

Arranging the Vowels

We can start by arranging the vowels in the odd places. Since there are three vowels and four odd places, we can arrange the vowels in 4P3 ways, which is equal to 4 x 3 x 2 = 24.

Arranging the Consonants

Once the vowels are arranged in the odd places, we can arrange the consonants in the even places. There are four consonants in the word DAUGHTER, which are D, G, H, and T. We can arrange these consonants in 4P4 ways, which is equal to 4 x 3 x 2 x 1 = 24.

Multiplying the Number of Arrangements

To determine the total number of different words that can be formed, we need to multiply the number of arrangements of the vowels and consonants.

24 (vowels) x 24 (consonants) = 576

Therefore, there are 576 different words that can be formed by arranging the letters of the word DAUGHTER so that the vowels occupy the odd places.

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