Forming Words Starting with 'D' from DELHI

The question asks for the number of distinct words that can be formed using all the letters of the word 'DELHI', with the condition that each word must start with the letter 'D'. All letters must be used, and repetition is not allowed.

Letters in the Word 'DELHI'

The word 'DELHI' has 5 unique letters:

• D
• E
• L
• H
• I

So, we have a total of 5 distinct letters to arrange.

Arrangement with Fixed Starting Letter

We need to form 5-letter words. The primary condition is that the word must begin with 'D'. This fixes the first position:

D _ _ _ _

Since 'D' is placed in the first position, we are left with the remaining 4 letters: {E, L, H, I}.

These 4 remaining letters need to be arranged in the remaining 4 positions (positions 2, 3, 4, and 5).

Calculating the Number of Arrangements

The number of ways to arrange 'n' distinct items is calculated using the factorial function, denoted as $n!$.

In this case, we need to arrange the 4 distinct letters {E, L, H, I} in the 4 available positions.

The number of possible arrangements is $4!$.

Calculation of $4!$: $$4! = 4 \times 3 \times 2 \times 1$$ $$4! = 12 \times 2 \times 1$$ $$4! = 24$$

Thus, there are 24 different ways to arrange the letters E, L, H, and I in the remaining four positions.

Conclusion

Each of these 24 arrangements corresponds to a unique word starting with 'D' formed using all the letters of 'DELHI'.

Therefore, the total number of words starting with the letter 'D' that can be formed is 24.