Find in how many different ways, the letters of the word ‘LEADING’ can be…

Step-by-Step Solution

1. Analyze the word: The word 'LEADING' has 7 unique letters: L, E, A, D, I, N, G.

2. Identify vowels and consonants:

   • Vowels: E, A, I (3 vowels)

   • Consonants: L, D, N, G (4 consonants)

3. Group the vowels: Since the vowels must always come together, we treat them as a single entity or 'block'. Let's call this block (EAI).

4. Count the items to arrange: Now, we arrange the 4 consonants (L, D, N, G) and the 1 vowel block (EAI). This gives us a total of 5 entities (4 + 1 = 5).

   • The number of ways to arrange these 5 entities is 5! (5 factorial).

   • 5! = 5 * 4 * 3 * 2 * 1 = 120 ways.

5. Arrange within the vowel block: The 3 vowels (E, A, I) inside their block can be arranged among themselves in 3! ways.

   • 3! = 3 * 2 * 1 = 6 ways.

6. Calculate the total arrangements: By the fundamental counting principle, we multiply these values:

   • Total ways = 120 * 6 = 720 ways.