Permutations from word DAUGHTER?

Question

1)	How many words can be formed using all 8 letters of word DAUGHTER with no repetition of letter allowed and vowels are located in odd position?  
2)	How many words can be formed using all 8 letters of word DAUGHTER with no repetition of letter allowed and vowels are located in even position?
The above two questions are to be solved as separate questions without actually tabulating and counting and should be simple enough to be explained to a student to whom the subject has introduced recently (say std x)
Also a small variation (please mention logic used)
3)	How many such 5 letter words can be created using letters of word DAUGHTER with no repetition of letter allowed and vowels are located in odd position?  
4)	How many such 5 letter words can be created using letters of word DAUGHTER with no repetition of letter allowed and vowels are located in even position?

M3 · Accepted Answer

q1
the 3 vowels can be permuted in the 4 odd positions in 4P3 ways, and the remaining consonants in 5P5 ways
ans: 4!*5! = 2880 <------

q2
do similarly

q3
this is tedious
Σ  (n vowels chosen & placed in one of 3 odd positions) * 
((5-n) consonants permuted in (5-n) positions), n =1 to 3
= 3C1*3P1 *5P4 + 3C2*3P2 *5P3 + 3C3*3P3 *5P2
= calculator ...... 

if 0 vowels are permitted, add 5P5 to the above

q4
do similarly
do similarly

? · Answer

How many WORDS or how many different PERMUTATIONS of letters?

If you are talking about WORDS, which is what you ask, then no one has yet given the correct answer.  To do so requires one to check to see if each different permutation is a WORD or not.

Using sowpods.txt as a reference (a word list with 267,751 words in it.  Google it if you haven't heard of it), there  are NO other words that use ALL 8 letters of the letters that make up the word DAUGHTER (and no other letters).

If one or more OTHER letters can also be used, then there are 61 different words that use ALL 8 of those letters.  Here's a portion of that list:  autographed. counterchanged, countercharged, dearbought, desulphurating. draughted, draughter, draughters, draughtswomen, fraughted, goddaughter, outcharged, overdraught, roughcasted, straughted, etc.

So as it is asked, the answer to #1 and #2 is zero.


#3: 5 is the answer.  The following five words are the only five-letter words that have vowels located in positions 1, 3, or 5, using the letters, d,a,u,g,h,t,e,r:  aruhe, grade, grate, trade, urate


#4: 16 is the answer.  The following sixteen words are the only five-letter words that have vowels located in positions 2 or 4, using the letters d,a,u,g,h,t,e,r:  dater, derat, derth, garth, gated, gater, gerah, hared, hated, hater, huger, raged, rahed, rated, retag, tared

This is a VERY odd question.  I can't believe I took the time to answer it.

? · Answer

1) Select a consonant (there are 5 choices). After selecting one, there are 4! ways to arrange the remaining consonants among themselves and 4! ways to arrange the 3 vowels and the chosen consonant among themselves. Therefore there are 5*4!*4! total possible permutations.
2) Identical logic as 1).
3) Vowels can be arranged 3! ways, times 5*4 arrangements of consonants gives a total of 3!*5*4=5! permutations.
4) Vowels arranged 3*2 ways, consonants arranged 5*4*3 ways, total of 3*2*5*4*3 ways.

Anonymous · Answer

1/
4 odd positions and 3 vowels ---> C(4,3) 3!5! = ...
2/ similar 1/

Anonymous · Answer

TUNAK TUNAK TUN TUNAK TUNAK TUN TUNAK TUNAK TUN DA DA DA !!!

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Permutations from word DAUGHTER?

5 Answers