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Given: The given number is 'GEOGRAPHY' Calculation: The word 'GEOGRAPHY' has 9 letters. It has the vowels E, O, A in it, and these 3 vowels must always come together. Hence these 3 vowels can be grouped and considered as a single letter. That is, GGRPHY(EOA). Let 7 letters in this word but in these 7 letters, 'G' occurs 2 times, but the rest of the letters are different. Now, The number of ways to arrange these letters = 7!/2! ⇒ 7 × 6 × 5 × 4 × 3 = 2520 In the 3 vowels(EOA), all vowels are different The number of ways to arrange these vowels = 3! ⇒ 3 × 2 × 1 = 6 Now, The required number of ways = 2520 × 6 ⇒ 15120 ∴ The required number of ways is 15120. India’s #1 Learning Platform Start Complete Exam Preparation
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In the below solved problem, every thing is okay, but if we have $4$ consonants then why we are giving $5!$? and is this a combination problem? how to distinguish? Question: In how many different ways can the letters of the word 'OPTICAL' be arranged so that the vowels always come together? Answer: The word 'OPTICAL' contains $7$ different letters. When the vowels OIA are always together, they can be supposed to form one letter. Then, we have to arrange the letters PTCL (OIA). Now, $5$ letters can be arranged in $5! = 120$ ways. The vowels (OIA) can be arranged among themselves in $3! = 6$ ways. Required number of ways $= (120*6) = 720$. $\endgroup$
Discussion :: Permutation and Combination - General Questions (Q.No.2)
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