In how many ways can the letters of the word, ‘language’ be arranged in such a way that the vowels always come together?

Hint: To solve this problem we have to know about the concept of permutations and combinations. But here a simple concept is used. In any given word, the number of ways we can arrange the word by jumbling the letters is the number of letters present in the word factorial. Here factorial of any number is the product of that number and all the numbers less than that number till 1. $ \Rightarrow n! = n(n - 1)(n - 2).......1$ Complete step by step answer: Given the word TRAINER, we have to arrange the letters of the word in such a way that all the vowels in the word TRAINER should be together. The number of vowels in the word TRAINER are = 3 vowels. The three vowels in the word TRAINER are A, I, and E. Now these three vowels should always be together and these vowels can be in any order, but they should be together. Here the three vowels AIE can be arranged in 3 factorial ways, as there are 3 vowels, as given below: The number of ways the 3 vowels AIE can be arranged is = $3!$ Now arranging the consonants other than the vowels is given by: As the left out letters in the word TRAINER are TRNR. The total no. of consonants left out are = 4 consonants. Now these 4 consonants can be arranged in the following way: As in the 4 letters TRNR, the letter R is repeated for 2 times, hence the letters TRNR can be arranged in : $ \Rightarrow \dfrac$ number of ways.

Topper’s approach: Total number of ways = 5! × 3! = 120 × 6 = 720 Detailed solution: To keep the vowels together we have to treat all the vowels as a single letter. ∴ Total no. of letters in ‘LEADING’ = 5 (L, D, N, G and the 3 vowels) ∴ 5 letters can be arranged in = 5! = 120 ways ∴ The 3 vowels can be arranged among themselves in = 3! = 6 ways ∴ Total number of ways = 120 × 6 = 720

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This problem has been solved! You'll get a detailed solution from a subject matter expert that helps you learn core concepts. See Answer See Answer See Answer done loading Question:In how many different ways can the letters of the word KNOWLEDGE be arranged in such a way that the vowels always come together?

The word 'LEADING' has 7 different letters. When the vowels EAI are always together, they can be supposed to form one letter. Then, we have to arrange the letters LNDG (EAI). Now, 5 (4 + 1 = 5) letters can be arranged in 5! = 120 ways. The vowels (EAI) can be arranged among themselves in 3! = 6 ways. Required number of ways = (120 x 6) = 720. In the word 'CORPORATION', we treat the vowels OOAIO as one letter. Thus, we have CRPRTN (OOAIO). This has 7 (6 + 1) letters of which R occurs 2 times and the rest are different. Number of ways arranging these letters = 7! = 2520. 2! Now, 5 vowels in which O occurs 3 times and the rest are different, can be arranged in 5! = 20 ways. 3! Required number of ways = (2520 x 20) = 50400. Number of ways of selecting (3 consonants out of 7) and (2 vowels out of 4) = ( 7C 3 x 4C 2) = 7 x 6 x 5 x 4 x 3 3 x 2 x 1 2 x 1 = 210. Number of groups, each having 3 consonants and 2 vowels = 210. Each group contains 5 letters. Number of ways of arranging 5 letters among themselves = 5! = 5 x 4 x 3 x 2 x 1 = 120. Required number of ways = (210 x 120) = 25200.