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?
Answers
In the word EQUATION, there are 5 vowels, namely, A, E, I, O, and U, and 3 consonants, namely, Q, T, and N.
Since all the vowels and consonants have to occur together, both (AEIOU) and (QTN) can be assumed as single objects. Then, the permutations of these 2 objects taken all at a time are counted. This number would be 2p2 = 2!
Corresponding to each of these permutations, there are 5! permutations of the five vowels taken all at a time and 3! permutations of the 3 consonants taken all at a time.
Hence, by multiplication principle, required number of words = 2! × 5! × 3!
= 1440
First of all, we have to separate the consonants and vowels and consider each time the set of the consonants and vowels as a single letter.
now,
here, word is E Q U A T I O N
vowels —> E , U, A , I , O ( there are five vowels in given words )
consonants—> T, Q , N ( there are 3 consonants in given words )
the vowels can be arranged in 5! ways
the consonants can be arranged in 3! ways
These vowels and consonants ( when we take as a single letter )can be arranged 2! ways .
hence, a/c to fundamental principle of counting
total number of ways = 5! × 3! × 2!
= (5×4 ×3 ×2) × (3 × 2) × (2 )
= 120 × 6 × 2
= 1440
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