How many words, with or without meaning, can be formed using all the letters of the word EQUATION at a time so that the vowels and consonants occur together?
The total number of words formed is 1440.
Explanation
The question asks for arrangements where vowels and consonants occur together. This is a grouping problem where we treat all vowels as one unit and all consonants as another unit. The context shows similar methodology in Example (ii) where objects are grouped together and then arranged internally.
Solution Steps
Step 1: Identify vowels and consonants in EQUATION. Vowels: E, U, A, I, O (5 vowels, all distinct) Consonants: Q, T, N (3 consonants, all distinct)
Step 2: Treat all vowels as one object and all consonants as another object. We have 2 objects which can be arranged in 2! = 2 ways.
Step 3: Arrange vowels within their group. The 5 vowels can be arranged in 5! = 120 ways.
Step 4: Arrange consonants within their group. The 3 consonants can be arranged in 3! = 6 ways.
Step 5: Apply multiplication principle. Total arrangements = 2! × 5! × 3! = × 6 = 1440