In how many different ways can the letters of the word 'LEADING' be arranged…

Step-by-Step Solution

To find the number of ways to arrange the letters of the word 'LEADING' such that the vowels always come together, follow these steps:

1. Identify the Letters:

   • The word 'LEADING' has 7 letters: L, E, A, D, I, N, G.

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

   • Vowels: E, A, I (3 letters)

2. Use the "Block" Method:

   • Since the vowels (E, A, I) must always come together, treat them as a single block: (EAI).

   • Now, you have the 4 consonants plus this 1 vowel block, making a total of 5 items to arrange: {L, D, N, G, (EAI)}.

3. Arrange the Items:

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

   • 5! = 5 * 4 * 3 * 2 * 1 = 120.

4. Arrange Within the Block:

   • The 3 vowels within the block (E, A, I) can also be arranged among themselves in 3! ways.

   • 3! = 3 * 2 * 1 = 6.

5. Calculate Total Arrangements:

   • Total arrangements = (Arrangements of items) * (Arrangements within vowel block)

   • Total = 120 * 6 = 720.