Teachers and Students sitting in a line with a condition
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A class consists of three teachers (Mr. P, Ms. Q, Ms. R) and six students (Ali, Bob, Cal, Dee, Amy, Fay). How many ways can they sit in a line of 9 chairs if we have the property that between any two teachers there are exactly two students?
The correct answer is supposed to be 12,960 ways, but I cannot determine this answer. My main assumption is that there are three teachers, the two of these students will have to sit alone on the sides of the teachers. Like so:
S T S S T S S T S
My attempt:
$6times3times5times4times2times3times2times1times1$ which equals $4320$.
combinatorics
asked Nov 26 at 18:40
Ludwigthestud
908
add a comment |
up vote
3
down vote
favorite
A class consists of three teachers (Mr. P, Ms. Q, Ms. R) and six students (Ali, Bob, Cal, Dee, Amy, Fay). How many ways can they sit in a line of 9 chairs if we have the property that between any two teachers there are exactly two students?
The correct answer is supposed to be 12,960 ways, but I cannot determine this answer. My main assumption is that there are three teachers, the two of these students will have to sit alone on the sides of the teachers. Like so:
S T S S T S S T S
My attempt:
$6times3times5times4times2times3times2times1times1$ which equals $4320$.
combinatorics
asked Nov 26 at 18:40
Ludwigthestud
908
2
You can also have SSTSSTSST or TSSTSSTSS.
– JMoravitz
Nov 26 at 18:41
Well, I'll be damned. My mind locked me into thinking that every two students needed a teacher to the left and right of them, but it was okay if there was lone students. I made the question harder than it needed to be.
– Ludwigthestud
Nov 26 at 18:45
2
@Ludwigthestud: that gives you exactly the missing factor $3$
– Ross Millikan
Nov 26 at 18:51
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
A class consists of three teachers (Mr. P, Ms. Q, Ms. R) and six students (Ali, Bob, Cal, Dee, Amy, Fay). How many ways can they sit in a line of 9 chairs if we have the property that between any two teachers there are exactly two students?
The correct answer is supposed to be 12,960 ways, but I cannot determine this answer. My main assumption is that there are three teachers, the two of these students will have to sit alone on the sides of the teachers. Like so:
S T S S T S S T S
My attempt:
$6times3times5times4times2times3times2times1times1$ which equals $4320$.
combinatorics
asked Nov 26 at 18:40
Ludwigthestud
908
A class consists of three teachers (Mr. P, Ms. Q, Ms. R) and six students (Ali, Bob, Cal, Dee, Amy, Fay). How many ways can they sit in a line of 9 chairs if we have the property that between any two teachers there are exactly two students?
The correct answer is supposed to be 12,960 ways, but I cannot determine this answer. My main assumption is that there are three teachers, the two of these students will have to sit alone on the sides of the teachers. Like so:
S T S S T S S T S
My attempt:
$6times3times5times4times2times3times2times1times1$ which equals $4320$.
combinatorics
combinatorics
asked Nov 26 at 18:40
Ludwigthestud
908
asked Nov 26 at 18:40
Ludwigthestud
908
asked Nov 26 at 18:40
Ludwigthestud
908
asked Nov 26 at 18:40
Ludwigthestud
908
asked Nov 26 at 18:40
Ludwigthestud
908
908
2
You can also have SSTSSTSST or TSSTSSTSS.
– JMoravitz
Nov 26 at 18:41
Well, I'll be damned. My mind locked me into thinking that every two students needed a teacher to the left and right of them, but it was okay if there was lone students. I made the question harder than it needed to be.
– Ludwigthestud
Nov 26 at 18:45
2
@Ludwigthestud: that gives you exactly the missing factor $3$
– Ross Millikan
Nov 26 at 18:51
add a comment |
2
You can also have SSTSSTSST or TSSTSSTSS.
– JMoravitz
Nov 26 at 18:41
Well, I'll be damned. My mind locked me into thinking that every two students needed a teacher to the left and right of them, but it was okay if there was lone students. I made the question harder than it needed to be.
– Ludwigthestud
Nov 26 at 18:45
2
@Ludwigthestud: that gives you exactly the missing factor $3$
– Ross Millikan
Nov 26 at 18:51
2
2
You can also have SSTSSTSST or TSSTSSTSS.
– JMoravitz
Nov 26 at 18:41
You can also have SSTSSTSST or TSSTSSTSS.
– JMoravitz
Nov 26 at 18:41
Well, I'll be damned. My mind locked me into thinking that every two students needed a teacher to the left and right of them, but it was okay if there was lone students. I made the question harder than it needed to be.
– Ludwigthestud
Nov 26 at 18:45
Well, I'll be damned. My mind locked me into thinking that every two students needed a teacher to the left and right of them, but it was okay if there was lone students. I made the question harder than it needed to be.
– Ludwigthestud
Nov 26 at 18:45
2
2
@Ludwigthestud: that gives you exactly the missing factor $3$
– Ross Millikan
Nov 26 at 18:51
@Ludwigthestud: that gives you exactly the missing factor $3$
– Ross Millikan
Nov 26 at 18:51
add a comment |
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2
You can also have SSTSSTSST or TSSTSSTSS.
– JMoravitz
Nov 26 at 18:41
Well, I'll be damned. My mind locked me into thinking that every two students needed a teacher to the left and right of them, but it was okay if there was lone students. I made the question harder than it needed to be.
– Ludwigthestud
Nov 26 at 18:45
2
@Ludwigthestud: that gives you exactly the missing factor $3$
– Ross Millikan
Nov 26 at 18:51