How many different words can be formed from the letters of the word DAUGHTER when the letters T,A,D are never together.
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How many different words can be formed from the letters of the word DAUGHTER when the letters T,A,D are never together.
combinatorics permutations
edited Aug 20 at 10:09
N. F. Taussig
38.7k93153
asked Aug 20 at 9:46
jame samajoe
569
add a comment |Â
up vote
-1
down vote
favorite
How many different words can be formed from the letters of the word DAUGHTER when the letters T,A,D are never together.
combinatorics permutations
edited Aug 20 at 10:09
N. F. Taussig
38.7k93153
asked Aug 20 at 9:46
jame samajoe
569
1
Please edit your question to show what you have attempted and explain where you are stuck so that you receive responses that address the specific difficulties you are encountering.
– N. F. Taussig
Aug 20 at 10:10
add a comment |Â
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
How many different words can be formed from the letters of the word DAUGHTER when the letters T,A,D are never together.
combinatorics permutations
edited Aug 20 at 10:09
N. F. Taussig
38.7k93153
asked Aug 20 at 9:46
jame samajoe
569
How many different words can be formed from the letters of the word DAUGHTER when the letters T,A,D are never together.
combinatorics permutations
edited Aug 20 at 10:09
N. F. Taussig
38.7k93153
asked Aug 20 at 9:46
jame samajoe
569
edited Aug 20 at 10:09
N. F. Taussig
38.7k93153
edited Aug 20 at 10:09
N. F. Taussig
38.7k93153
edited Aug 20 at 10:09
N. F. Taussig
38.7k93153
38.7k93153
asked Aug 20 at 9:46
jame samajoe
569
asked Aug 20 at 9:46
jame samajoe
569
asked Aug 20 at 9:46
jame samajoe
569
569
1
Please edit your question to show what you have attempted and explain where you are stuck so that you receive responses that address the specific difficulties you are encountering.
– N. F. Taussig
Aug 20 at 10:10
add a comment |Â
1
Please edit your question to show what you have attempted and explain where you are stuck so that you receive responses that address the specific difficulties you are encountering.
– N. F. Taussig
Aug 20 at 10:10
1
1
Please edit your question to show what you have attempted and explain where you are stuck so that you receive responses that address the specific difficulties you are encountering.
– N. F. Taussig
Aug 20 at 10:10
Please edit your question to show what you have attempted and explain where you are stuck so that you receive responses that address the specific difficulties you are encountering.
– N. F. Taussig
Aug 20 at 10:10
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
1
down vote
accepted
I got another answer,it's pretty easy to know that the total of words you can form are $8!$, without any condition. Then, suppose the letters $TAD$ are together and they're one with the $5$ left. Then you have $6!$ ways to write words with $TAD$ together. This is because you can arrange $TAD$ on any of the six spaces, and then the other letters will have $5!$ ways to arrange, then $6*5!=6!$ $$(TAD)UGHER$$$$UG(TAD)HRE$$$$...$$
Remember that you can arrange $TAD$ as $TDA, ADT, ATD, DTA$ and $DAT$, so you have $6!*6$ words with the letters $T,A,D$ together. So you just need to take them away:
$$8!-6!*6=40320-4320=36000$$
Hence the answer is $36000$
answered Aug 21 at 18:21
cptnSonoda
204110
add a comment |Â
up vote
2
down vote
It is obvious that
-The words can be formed with that letters without any assumption are $8!$
-The letters $T,A,D$ together, can be written with $6$ way $(TAD,TDA,ATD,...)$
We can see that each one of the previous six possibilities can appear in $6$ position, such that the other $5$ letters can be arranged in $5!$ way , so all of them appear $6times 6 times 5!$ time, and the answer is ($8!-6times6!$)
answered Aug 20 at 10:18
The_lost
1,067318
2
You forgot that every of those 36 times they appear, the other 5 letters can be arranged too
– cptnSonoda
Aug 21 at 18:23
@cptnSonoda thank you, I have fixed it.
– The_lost
Aug 22 at 0:06
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
I got another answer,it's pretty easy to know that the total of words you can form are $8!$, without any condition. Then, suppose the letters $TAD$ are together and they're one with the $5$ left. Then you have $6!$ ways to write words with $TAD$ together. This is because you can arrange $TAD$ on any of the six spaces, and then the other letters will have $5!$ ways to arrange, then $6*5!=6!$ $$(TAD)UGHER$$$$UG(TAD)HRE$$$$...$$
Remember that you can arrange $TAD$ as $TDA, ADT, ATD, DTA$ and $DAT$, so you have $6!*6$ words with the letters $T,A,D$ together. So you just need to take them away:
$$8!-6!*6=40320-4320=36000$$
Hence the answer is $36000$
answered Aug 21 at 18:21
cptnSonoda
204110
add a comment |Â
up vote
1
down vote
accepted
I got another answer,it's pretty easy to know that the total of words you can form are $8!$, without any condition. Then, suppose the letters $TAD$ are together and they're one with the $5$ left. Then you have $6!$ ways to write words with $TAD$ together. This is because you can arrange $TAD$ on any of the six spaces, and then the other letters will have $5!$ ways to arrange, then $6*5!=6!$ $$(TAD)UGHER$$$$UG(TAD)HRE$$$$...$$
Remember that you can arrange $TAD$ as $TDA, ADT, ATD, DTA$ and $DAT$, so you have $6!*6$ words with the letters $T,A,D$ together. So you just need to take them away:
$$8!-6!*6=40320-4320=36000$$
Hence the answer is $36000$
answered Aug 21 at 18:21
cptnSonoda
204110
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
I got another answer,it's pretty easy to know that the total of words you can form are $8!$, without any condition. Then, suppose the letters $TAD$ are together and they're one with the $5$ left. Then you have $6!$ ways to write words with $TAD$ together. This is because you can arrange $TAD$ on any of the six spaces, and then the other letters will have $5!$ ways to arrange, then $6*5!=6!$ $$(TAD)UGHER$$$$UG(TAD)HRE$$$$...$$
Remember that you can arrange $TAD$ as $TDA, ADT, ATD, DTA$ and $DAT$, so you have $6!*6$ words with the letters $T,A,D$ together. So you just need to take them away:
$$8!-6!*6=40320-4320=36000$$
Hence the answer is $36000$
answered Aug 21 at 18:21
cptnSonoda
204110
I got another answer,it's pretty easy to know that the total of words you can form are $8!$, without any condition. Then, suppose the letters $TAD$ are together and they're one with the $5$ left. Then you have $6!$ ways to write words with $TAD$ together. This is because you can arrange $TAD$ on any of the six spaces, and then the other letters will have $5!$ ways to arrange, then $6*5!=6!$ $$(TAD)UGHER$$$$UG(TAD)HRE$$$$...$$
Remember that you can arrange $TAD$ as $TDA, ADT, ATD, DTA$ and $DAT$, so you have $6!*6$ words with the letters $T,A,D$ together. So you just need to take them away:
$$8!-6!*6=40320-4320=36000$$
Hence the answer is $36000$
answered Aug 21 at 18:21
cptnSonoda
204110
answered Aug 21 at 18:21
cptnSonoda
204110
answered Aug 21 at 18:21
cptnSonoda
204110
answered Aug 21 at 18:21
cptnSonoda
204110
204110
add a comment |Â
add a comment |Â
up vote
2
down vote
It is obvious that
-The words can be formed with that letters without any assumption are $8!$
-The letters $T,A,D$ together, can be written with $6$ way $(TAD,TDA,ATD,...)$
We can see that each one of the previous six possibilities can appear in $6$ position, such that the other $5$ letters can be arranged in $5!$ way , so all of them appear $6times 6 times 5!$ time, and the answer is ($8!-6times6!$)
answered Aug 20 at 10:18
The_lost
1,067318
2
You forgot that every of those 36 times they appear, the other 5 letters can be arranged too
– cptnSonoda
Aug 21 at 18:23
@cptnSonoda thank you, I have fixed it.
– The_lost
Aug 22 at 0:06
add a comment |Â
up vote
2
down vote
It is obvious that
-The words can be formed with that letters without any assumption are $8!$
-The letters $T,A,D$ together, can be written with $6$ way $(TAD,TDA,ATD,...)$
We can see that each one of the previous six possibilities can appear in $6$ position, such that the other $5$ letters can be arranged in $5!$ way , so all of them appear $6times 6 times 5!$ time, and the answer is ($8!-6times6!$)
answered Aug 20 at 10:18
The_lost
1,067318
2
You forgot that every of those 36 times they appear, the other 5 letters can be arranged too
– cptnSonoda
Aug 21 at 18:23
@cptnSonoda thank you, I have fixed it.
– The_lost
Aug 22 at 0:06
add a comment |Â
up vote
2
down vote
up vote
2
down vote
It is obvious that
-The words can be formed with that letters without any assumption are $8!$
-The letters $T,A,D$ together, can be written with $6$ way $(TAD,TDA,ATD,...)$
We can see that each one of the previous six possibilities can appear in $6$ position, such that the other $5$ letters can be arranged in $5!$ way , so all of them appear $6times 6 times 5!$ time, and the answer is ($8!-6times6!$)
answered Aug 20 at 10:18
The_lost
1,067318
It is obvious that
-The words can be formed with that letters without any assumption are $8!$
-The letters $T,A,D$ together, can be written with $6$ way $(TAD,TDA,ATD,...)$
We can see that each one of the previous six possibilities can appear in $6$ position, such that the other $5$ letters can be arranged in $5!$ way , so all of them appear $6times 6 times 5!$ time, and the answer is ($8!-6times6!$)
answered Aug 20 at 10:18
The_lost
1,067318
edited Aug 22 at 0:05
answered Aug 20 at 10:18
The_lost
1,067318
answered Aug 20 at 10:18
The_lost
1,067318
answered Aug 20 at 10:18
The_lost
1,067318
1,067318
2
You forgot that every of those 36 times they appear, the other 5 letters can be arranged too
– cptnSonoda
Aug 21 at 18:23
@cptnSonoda thank you, I have fixed it.
– The_lost
Aug 22 at 0:06
add a comment |Â
2
You forgot that every of those 36 times they appear, the other 5 letters can be arranged too
– cptnSonoda
Aug 21 at 18:23
@cptnSonoda thank you, I have fixed it.
– The_lost
Aug 22 at 0:06
2
2
You forgot that every of those 36 times they appear, the other 5 letters can be arranged too
– cptnSonoda
Aug 21 at 18:23
You forgot that every of those 36 times they appear, the other 5 letters can be arranged too
– cptnSonoda
Aug 21 at 18:23
@cptnSonoda thank you, I have fixed it.
– The_lost
Aug 22 at 0:06
@cptnSonoda thank you, I have fixed it.
– The_lost
Aug 22 at 0:06
add a comment |Â
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1
Please edit your question to show what you have attempted and explain where you are stuck so that you receive responses that address the specific difficulties you are encountering.
– N. F. Taussig
Aug 20 at 10:10