In how many ways can we arrange the word coffeehouse so that all the vowels come together?
Answer Show
Verified Hint: Here, we are required to arrange the letters in the given word ‘FACTOR’. Thus, we will use Permutations to ‘arrange’ the letters keeping in mind that all the letters in the given word are unique. Thus, applying the formula and solving the factorial, we will be able to find the required ways of arrangement of letters of the given word. Formula Used: Complete step-by-step answer: Therefore, we can arrange the letters in the word ‘FACTOR’ in 720 ways. Note: In how many different ways can the letters of the word TRAINER be arranged so that the vowels always come together?A. 1440B. 120C. 720D. 360Answer Verified
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. Complete step by step answer: The number of ways the word TRAINER can be arranged so that the vowels always come together are 360. Note: Here while solving such kind of problems if there is any word of $n$ letters and a letter is repeating for $r$ times in it, then it can be arranged in $\dfrac{{n!}}{{r!}}$ number of ways. If there are many letters repeating for a distinct number of times, such as a word of $n$ letters and ${r_1}$ repeated items, ${r_2}$ repeated items,…….${r_k}$ repeated items, then it is arranged in $\dfrac{{n!}}{{{r_1}!{r_2}!......{r_k}!}}$ number of ways. How many ways so that vowels come together?The number of ways the word TRAINER can be arranged so that the vowels always come together are 360. Note: Here while solving such kind of problems if there is any word of n letters and a letter is repeating for r times in it, then it can be arranged in n! r!
How many ways can vowels be arranged?The vowels (EAI) can be arranged among themselves in 3! = 6 ways. Required number of ways = (120 x 6) = 720.
How many ways a word can be arranged so that vowels never come together?number of arrangements in which the vowels do not come together =5040−1440=3600 ways.
How many arrangements of leading are possible where all vowels are together?Solution(By Examveda Team)
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.
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