How many arrangements of the word daughter are there if none of the vowels can ever be together?
Solution : No. of letters in the word 'DAUGHTER' =8 Show
Hi all, I would appreciate any feedback on my solution to this problem. Problem: How many arrangements of the word DAUGHTER are there if none of the vowels can ever be together? Solution attempt: 3 vowels and 5 consonants Arrangements with no restrictions = 8! = 40320All (three) vowels together = 6!3!=4320Two vowels together = 7!2!=10080 So, No vowels together = No restrictions - three vowels together - two vowels together=40320 - 4320 - 10080 = 25920 ways Does my logic make sense? Have I missed any cases? I'm trying to use permutations only. Thanks for the help! Edit: Thanks for all the help! The gist I was getting from the comments was that when I subtract 2 vowels together I am also getting 'more than 2 vowels together' (so, three vowels). To compensate for this I should add the ways of arranging 3 vowels together back so as to not subtract twice. No restrictions - 2 vowels together + 3 vowels together = 8! - 7! * 6 + 6!3! = 14400 How many words can be formed from the letters of the word ‘DAUGHTER’ so that(i) The vowels always come together?(ii) The vowels never come together?Answer Verified
Hint: The word daughter has $8$ letters in which $3$ are vowels. For the vowels to always come together consider all the $3$ vowels to be one letter (suppose V) then total letters become $6$ which can be arranged in $6!$ ways and the vowels themselves in $3!$ ways.Complete step-by-step answer: (ii)We have to find the number of words formed when no vowels are together. Note: Combination is used when things are to be arranged but not necessarily in order. Permutation is a little different. In permutation, order is important. Permutation is given by- How many words can be formed from daughter if vowels are not together?The total number of words formed from 'DAUGHTER' such that no vowels are together is 14400.
How many possible arrangements of daughter when vowels come together?∴ Required number of words = 6 ! × 3 ! = 6 × 5 × 4 × 3 × 2 × 1 × 3 × 2 = 4320.
How many ways can the letters of the word daughter be arranged to that the vowels may appear in the odd places?So, there are 4 odd places. Hence, there are 2880 different words.
How many ways Word arrange can be arranged in which vowels are not together?number of arrangements in which the vowels do not come together =5040−1440=3600 ways.
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