Arranging the word "LEADING" with All Vowels Together

To solve this problem, we need to arrange the letters of the word "LEADING" in such a way that all the vowels come together. Here are the steps we can follow:

Step 1: Identify the Vowels and Consonants

The word "LEADING" has three vowels (E, A, and I) and four consonants (L, D, N, and G).

Step 2: Treat the Vowels as a Group

Since we want all the vowels to come together, we can treat them as a group and arrange them as a single unit. This means that we can consider the group of vowels (EAI) as a single letter.

Step 3: Permute the Consonants

After treating the vowels as a group, we now have two letters to arrange: the vowel group (EAI) and the four consonants (L, D, N, and G). We can arrange these two letters in (2+4)! = 720 ways.

Step 4: Permute the Vowel Group

Within the vowel group (EAI), there are 3! = 6 ways to arrange the vowels. This means that we can arrange the letters in the vowel group in 6 different ways.

Step 5: Compute the Total Number of Arrangements

To get the total number of arrangements, we need to multiply the number of ways to permute the consonants by the number of ways to permute the vowel group. This gives us:

720 x 6 = 4,320

Therefore, there are 4,320 different ways to arrange the letters of the word "LEADING" such that all the vowels come together.

Conclusion:

In conclusion, we can arrange the letters of the word "LEADING" in 4,320 different ways such that all the vowels come together. We can follow the steps mentioned above to arrive at this answer.

3!×5!