How many different ways can the letter of the word COMPUTER be arranged so that vowels always come together?
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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 
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. There are 8 different letters
in the word 'COMPUTER' among them three are vowels and 5 are consonants. We want to arrange all the letters at a time such that: a) All the vowels are always together. It means that we cant arrange three vowels separately with 5 consonants. So, we assume 3 vowels as a single vowel with respect to 5 consonants. Thus, there are all together 6 letters and they can be arranged by 6! ways. But 3 vowels can be arranged to each other by 3! ways. Hence,
By multiplication principle of counting:- Total required ways = 6! * 3! = 4320 b) Vowels may occupy only an odd position. Clearly, there are 4 odd positions and 3 vowels. Thus, 3 vowels can be arranged in 4 odd positions by P(4,3) ways. After arranging 3 vowels in an odd position, we arrange 5 consonants in the remaining 5 seats by 5! ways. Hence, By multiplication principle of counting:- Total required ways = P(4,3) *
5! = 24*120 = 2880 c) Relative path of vowels and consonants are not changed. It means, we can't arrange vowels and consonants separately but we arrange vowels with vowels and consonants with consonants. Thus, 5 consonants can be arranged by 5! ways, and 3 vowels can be arranged by 3! ways. Hence, By multiplication principle of counting:- Total required ways = 5! * 3! = 720 Thus in this way, the word COMPUTER can be
arranged if the above conditions are given.
Answer: Option D Nội dung chính
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Explanation: The word 'MATHEMATICS' has 11 letters. It has the vowels 'A','E','A','I' in it and these 4 vowels must always come together. Hence these 4 vowels can be grouped and considered as a single letter. That is, MTHMTCS(AEAI). Hence we can assume total letters as 8. But in these 8 letters, 'M' occurs 2 times, 'T' occurs 2 times but rest of the letters are different. Hence,number of ways to arrange these letters In the 4 vowels (AEAI), 'A' occurs 2 times and rest of the vowels are different. Number of ways to arrange these vowels among themselves $=\dfrac{4!}{2!}=\dfrac{4×3×2×1}{2×1}=12$ Hence, required number of ways kanchan 2015-10-22 10:47:46 in how many times both T donot come together jiju 2015-10-23 13:47:13 to find that, remove both 'T's. Then we have 9 letters out of which 'A' occurs 2 times and 'M' occurs 2 times. Now there are 10 available positions and 2 positions can be selected from them in 10C2 = 45 ways The two 'E's can be arranged in the selected 2 positions only in 1 way. Therefore, number of arrangements where both 'T's do not come together = 90720 × 45 (use Q&A for new questions) Name Transcribed Image Text:In how many different ways can the letters of the word 'MATHEMATICS' be arranged such that the vowels must always come together? * 9800 O 100020 O 140020 O 120960 Q: Three balls are selected at random from a bag containing 2 red, 4 orange and 3 green balls. What is ... A: Given,no.of red balls=2no.of orange balls=4no.of green balls=3Total no.of balls=9 Q: Below is an incomplete probability distribution for a random variable X. 1 2 3 4 5 6 |f(x) |f(x) | 0... A: As per our guidelines we can solve first three sub part of question and rest can be reposted. Soluti... Q: Dora recently explored southeast Australia where she found some very large trees known as Eucalyptus... 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A: Given that The probability of the first one hitting the target=P(A) =1/4 The probability of the sec... How many ways can the letters in MATHEMATICS be arranged?There are 24 different ways to arrage the letters in the word math . How many vowels are there in MATHEMATICS?In the word 'Mathematics', we treat the vowels A, E, A, I as one letter. Thus, we have MTHMTCS (AEAI). Now, we have to arrange letters, out of which M occurs twice, T occurs twice, and the rest are different. Now, vowels A, E, I, A, has 4 letters in which A occurs 2 times and rest are different. How many ways to arrange the letters in MATHEMATICS if the vowels are always together at the end of word?120960 ways . Originally Answered: In how many different ways can the letters of the word mathematics be arranged so that the vowels always come together? How many arrangements can be made with the letters of the word MATHEMATICS order of vowels remains unchanged?The answer is 10,080 ways. How many vowels are in the word computer?There are 3 vowels and 6(5 consonant units + 1 vowel unit) objects that can be rearranged. How many different way can be formed by using all the letters of the word computer so that the vowels always come together?Solution(By Examveda Team) The given word contains 8 different letters. We keep the vowels (OAE) together and treat them as 1 letter. The vowels (OAE) can be arranged among themselves in 3! = 6 ways. How many ways can the letters of the word machine be arranged so that the vowels may occupy only the odd positions?Required number of ways = (360 * 2) = 720. In how many different ways can the letters of the word 'MACHINE' be arranged so that the vowels may occupy only the odd positions ? Now, 3 vowels can be placed at any of the three places, out of the four marked 1, 3, 5,7. How many arrangements of the letters of the word computer are possible if they have co together in order?a. All the eight letters in the word COMPUTER are distinct, so the number of ways in which we can arrange the letters equals the number of permutations of a set of eight elements. This equals 8! = 40,320. How many vowels are in the word computer?In the given word, there are 3 vowels O, U, and E which will be treated as 1 unit.
How many ways can the letters in the word computer be arranged if the letters CO must remain next to each other in order as a unit?c. When the letters are arranged randomly in a row, the total number of arrangements is 40,320 by part (a), and the number of arrangements with the letters CO next to each other (in order) as a unit is 5,040.
How many ways can you arrange the vowels?The vowels (EAI) can be arranged among themselves in 3! = 6 ways. Required number of ways = (120 x 6) = 720.
How many ways can the letters of the word machine be arranged so that the vowels may occupy only the odd positions?Required number of ways = (360 * 2) = 720. In how many different ways can the letters of the word 'MACHINE' be arranged so that the vowels may occupy only the odd positions ? Now, 3 vowels can be placed at any of the three places, out of the four marked 1, 3, 5,7.
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