Ten books are to be arranged on a shelf. Permutation Question
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Ten books are to be arranged on a shelf. Three of the books must be together (in any order)because they form a trilogy, and another two must be together in the correct order (volume 1,then volume 2). In how many ways can the books be arranged?
This is a question from my textbook and the answer says 51,840 which I do not know how they get.
My working is that, I consider the trilogy as one book and the pair of books also as one. Thus I thought that we only need to find how many ways we can arrange the 5 remaining books and multiply it with the number of ways we can get for the trilogy and the pair. Since the order does not matter for the trilogy, I used $3!$ and I said that it does not matter where the pair goes.
Thus my calculation was $5! times 3! =720$ which is very off to the answer. Any help would be appreciated!
combinatorics permutations combinations
asked Sep 6 at 11:49
user415903
193
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up vote
1
down vote
favorite
Ten books are to be arranged on a shelf. Three of the books must be together (in any order)because they form a trilogy, and another two must be together in the correct order (volume 1,then volume 2). In how many ways can the books be arranged?
This is a question from my textbook and the answer says 51,840 which I do not know how they get.
My working is that, I consider the trilogy as one book and the pair of books also as one. Thus I thought that we only need to find how many ways we can arrange the 5 remaining books and multiply it with the number of ways we can get for the trilogy and the pair. Since the order does not matter for the trilogy, I used $3!$ and I said that it does not matter where the pair goes.
Thus my calculation was $5! times 3! =720$ which is very off to the answer. Any help would be appreciated!
combinatorics permutations combinations
asked Sep 6 at 11:49
user415903
193
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Ten books are to be arranged on a shelf. Three of the books must be together (in any order)because they form a trilogy, and another two must be together in the correct order (volume 1,then volume 2). In how many ways can the books be arranged?
This is a question from my textbook and the answer says 51,840 which I do not know how they get.
My working is that, I consider the trilogy as one book and the pair of books also as one. Thus I thought that we only need to find how many ways we can arrange the 5 remaining books and multiply it with the number of ways we can get for the trilogy and the pair. Since the order does not matter for the trilogy, I used $3!$ and I said that it does not matter where the pair goes.
Thus my calculation was $5! times 3! =720$ which is very off to the answer. Any help would be appreciated!
combinatorics permutations combinations
asked Sep 6 at 11:49
user415903
193
Ten books are to be arranged on a shelf. Three of the books must be together (in any order)because they form a trilogy, and another two must be together in the correct order (volume 1,then volume 2). In how many ways can the books be arranged?
This is a question from my textbook and the answer says 51,840 which I do not know how they get.
My working is that, I consider the trilogy as one book and the pair of books also as one. Thus I thought that we only need to find how many ways we can arrange the 5 remaining books and multiply it with the number of ways we can get for the trilogy and the pair. Since the order does not matter for the trilogy, I used $3!$ and I said that it does not matter where the pair goes.
Thus my calculation was $5! times 3! =720$ which is very off to the answer. Any help would be appreciated!
combinatorics permutations combinations
combinatorics permutations combinations
asked Sep 6 at 11:49
user415903
193
asked Sep 6 at 11:49
user415903
193
asked Sep 6 at 11:49
user415903
193
asked Sep 6 at 11:49
user415903
193
asked Sep 6 at 11:49
user415903
193
193
add a comment |Â
add a comment |Â
1 Answer
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oldest
votes
up vote
7
down vote
Your textbook answer seems off too:
- $5$ books + $1$ pack of three + $1$ pair: $(5+1+1)! =7!$ arrangements
- the pack of three: $3!$ arrangements within the pack
- the pair has fixed order: 1 arrangement
All together:
$$7! cdot 3! = 30240$$
answered Sep 6 at 12:02
trancelocation
5,7701515
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
7
down vote
Your textbook answer seems off too:
- $5$ books + $1$ pack of three + $1$ pair: $(5+1+1)! =7!$ arrangements
- the pack of three: $3!$ arrangements within the pack
- the pair has fixed order: 1 arrangement
All together:
$$7! cdot 3! = 30240$$
answered Sep 6 at 12:02
trancelocation
5,7701515
add a comment |Â
up vote
7
down vote
Your textbook answer seems off too:
- $5$ books + $1$ pack of three + $1$ pair: $(5+1+1)! =7!$ arrangements
- the pack of three: $3!$ arrangements within the pack
- the pair has fixed order: 1 arrangement
All together:
$$7! cdot 3! = 30240$$
answered Sep 6 at 12:02
trancelocation
5,7701515
add a comment |Â
up vote
7
down vote
up vote
7
down vote
Your textbook answer seems off too:
- $5$ books + $1$ pack of three + $1$ pair: $(5+1+1)! =7!$ arrangements
- the pack of three: $3!$ arrangements within the pack
- the pair has fixed order: 1 arrangement
All together:
$$7! cdot 3! = 30240$$
answered Sep 6 at 12:02
trancelocation
5,7701515
Your textbook answer seems off too:
- $5$ books + $1$ pack of three + $1$ pair: $(5+1+1)! =7!$ arrangements
- the pack of three: $3!$ arrangements within the pack
- the pair has fixed order: 1 arrangement
All together:
$$7! cdot 3! = 30240$$
answered Sep 6 at 12:02
trancelocation
5,7701515
answered Sep 6 at 12:02
trancelocation
5,7701515
answered Sep 6 at 12:02
trancelocation
5,7701515
answered Sep 6 at 12:02
trancelocation
5,7701515
5,7701515
add a comment |Â
add a comment |Â
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