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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 arrangements can be made using the word TESTBOOK so that all the vowels come together
Answer (Detailed Solution Below)Option 5 : None of the above/More than one of the above Shortcut Trick Consider all the vowels together as one unit like TSTBK (EOO) Now, total number of units is (5 + 1) = 6. the units can arrange themselves in 6! ways . the vowels (EOO) can arrange themselves in 3! ways . Repeating letters = O & T So, the total number of ways = (6! × 3!)/2! × 2! = 1080 How many arrangements vowels occur together?` <br> Hence, the number of arrangement in which 4 vowels are together `=(10080 xx 12)=120960. `
How many arrangements can be formed by the letters of the word vowels?=3×2×4×3×2=144.
How many arrangements can be made of the letters of the word arrangement?=360 the number of ways.
How many ways can the letters of the word ideal be arranged so that all the consonants are always together?∴ The number of ways of arranging these letter is 64800.
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