Question

No of words formed by the letters of the word HINDUSTA which begin and end in a vowel are?

Answer 1

4320

Answer 2

To form words from the letters of HINDUSTA that begin and end in a vowel:

1. Identify Letters and Categories:
   The word HINDUSTA has 8 distinct letters: H, I, N, D, U, S, T, A.
   Vowels: I, U, A (3 vowels)
   Consonants: H, N, D, S, T (5 consonants)

2. Place the First Vowel:
   The first position of the word must be a vowel. There are 3 choices for this position (I, U, or A).
   Number of ways to fill the first position = 3.

3. Place the Last Vowel:
   The last position of the word must also be a vowel. Since one vowel has already been placed in the first position, there are 2 vowels remaining.
   Number of ways to fill the last position = 2.

4. Arrange the Remaining Letters:
   After placing two vowels at the beginning and end, there are 8 - 2 = 6 letters remaining. These 6 letters are the 5 consonants and the 1 vowel that was not used. All these 6 letters are distinct.
   There are 6 remaining positions in the middle of the word. These 6 distinct letters can be arranged in these 6 positions in 6! ways.
   6! = 6 × 5 × 4 × 3 × 2 × 1 = 720.

5. Calculate the Total Number of Words:
   To find the total number of words satisfying the conditions, multiply the number of ways for each step:
   Total words = (Ways to choose first vowel) × (Ways to choose last vowel) × (Ways to arrange remaining letters)
   Total words = 3 × 2 × 6!
   Total words = 6 × 720
   Total words = 4320