How many different words can be formed from the word daughter so that ending and beginning letters are Consonents?

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The given word, 'DAUGHTER' contains 3 vowels A, U, E and 5 consonants, D, G, H, T, R.Case (i) When all vowels occur together :Let us assume (AUE) as a single letter.Then, this letter (AUE) along with 5 other letters can be arranged in 6P6=(6!) ways = (6×5×4×3×2×1) ways= 720 ways.These 3 vowels may be arranged among themselves in 3 !=6 ways.Hence, the required number of words with vowels together= (6 !)×(3 !)=(720×6)=4320.Case (ii) When all vowels do not occur together.Number of words formed by using all the 8 letters of the given word= 8P8=8 !=(8×7×6×5×4×3×2×1)=40320.Number of words in which all vowels are never together = (total number of words) - (number of words with all vowels together)=(40320−4320)=36000.
How many 4 letter word can be formed from the given word daughter such that every word must contain the letter G?
How many words can you make out of daughter?

Solution

The given word, 'DAUGHTER' contains 3 vowels A, U, E and 5 consonants, D, G, H, T, R.Case (i) When all vowels occur together :Let us assume (AUE) as a single letter.Then, this letter (AUE) along with 5 other letters can be arranged in 6P6=(6!) ways = (6×5×4×3×2×1) ways= 720 ways.These 3 vowels may be arranged among themselves in 3 !=6 ways.Hence, the required number of words with vowels together= (6 !)×(3 !)=(720×6)=4320.Case (ii) When all vowels do not occur together.Number of words formed by using all the 8 letters of the given word= 8P8=8 !=(8×7×6×5×4×3×2×1)=40320.Number of words in which all vowels are never together = (total number of words) - (number of words with all vowels together)=(40320−4320)=36000.

Solution : The letters of the word daughter are “d,a,u,g,h,t,e,r”. So, the vowels are ‘a, u, e’ and the consonants are “d,g,h,t,r”. (i)Now, all the vowels should come together, so consider the bundle of vowels as one letter, then total letters will be `6`. So, the number of words formed by these letters will be `6!` but, the vowels can be arranged differently in the bundle, resulting in different words, so we have to consider the arrangements of the `3` vowels. So, the arrangements of vowels will be `3!` Thus, the total number of words formed will be equal to `(6!×3!)=4320` (ii)First arrange `5` consonants in five places in `5!` ways. `6` gaps are created. Out of these `6` gaps, select `3` gaps in `6_(C_3)` ways and then make the vowels permute in those `3` selected places in `3!` ways. This leads `5!×6_(C_3)xx3!`=14400.

Solution : No. of letters in the word 'DAUGHTER' =8 (i) If there is no restriction, No. of permutations to fill 8 places with given 8 letters `= P_(8) = 8! = 40320`. (ii) In the word 'DAUGHTER'. Vowels = A,U,E and Consonants = D,G,T,H,R `rArr` No. of vowels and consonants are 3 and 5 respectively. `:'` A,U,E are always together `:.` Treating these letters as one letter, the letters are `5 +1 = 6` Now the number of arrangement of 6 letters at 6 places `= .^(6)P_(6) = 6! = 720` and Number of arrangement of letters A,U,E at 3 places `= .^(3)P_(3) = 3! =6`. `:.` Required arrangements `= 720 xx 6 = 4320` (iii) For the words starting with A A will be at first place and no of arrangements of remaining 7 letters at remaining 7 places `=.^(7)P_(7) = 7! = 5040`. (iv) For the words starting with A and ending with R, No. of arrangements of remaining 6 letters at remaining 6 places `=.^(6)P_(6) = 6! = 720`.