Question: Number of different words that can be formed using all the letters of the word "DEEPMALA" if two vow...

Question

Number of different words that can be formed using all the letters of the word "DEEPMALA" if two vowels are together and the other two are also together but separated from the first two is

Answer 1

Solution

Here's a breakdown of the solution:

Identify Letters: The word DEEPMALA has 8 letters: D, P, M, L (4 consonants) and E, E, A, A (4 vowels).
Condition Breakdown: We need to form two groups of two vowels each ( $G_1$ and $G_2$ ) such that they are separated.
Form Vowel Groups:
- Case 1: Groups are (EE) and (AA).
  - The block (EE) has 1 internal arrangement. The block (AA) has 1 internal arrangement.
  - These two blocks are distinct.
  - Arrange the 4 consonants (D, P, M, L) in $4! = 24$ ways.
  - This creates 5 slots for the vowel blocks.
  - Place the 2 distinct blocks ((EE), (AA)) into 2 of these 5 slots in $P(5, 2) = 20$ ways.
  - Total for Case 1 = $24 \times 20 = 480$ .
- Case 2: Groups are (EA) and (EA).
  - Each (EA) group can be arranged internally as (EA) or (AE) ( $2! = 2$ ways).
  - This leads to three sub-scenarios for the actual blocks formed:
    - Subcase 2a: Both blocks are (EA) and (EA). The blocks are identical.
      - Arrange consonants: $4! = 24$ ways.
      - Place 2 identical (EA) blocks in 5 slots: $C(5, 2) = 10$ ways.
      - Total = $24 \times 10 = 240$ .
    - Subcase 2b: Both blocks are (AE) and (AE). The blocks are identical.
      - Arrange consonants: $4! = 24$ ways.
      - Place 2 identical (AE) blocks in 5 slots: $C(5, 2) = 10$ ways.
      - Total = $24 \times 10 = 240$ .
    - Subcase 2c: One block is (EA) and the other is (AE). The blocks are distinct.
      - Arrange consonants: $4! = 24$ ways.
      - Place 2 distinct blocks ((EA), (AE)) in 5 slots: $P(5, 2) = 20$ ways.
      - Total = $24 \times 20 = 480$ .
  - Total for Case 2 = $240 + 240 + 480 = 960$ .
Total Words: Sum of words from Case 1 and Case 2 = $480 + 960 = 1440$ .