- #1
recon
- 401
- 1
There are 9 different books arranged neatly on a shelf. 4 of them have labels on them. In how many ways can the books be arranged such that the labelled books are separated from each other?
Here's my sorry attempt at trying to solve the problem:
There are only 12 different configurations of 'slots' into which we can put the labelled books so that they are separated. This is shown below. The slots are represented by 'x' below. The unlabelled books are represented by 'u'.
1st configuration - x,u,x,u,x,u,x,u,u
2nd configuration - u,x,u,x,u,x,u,x,u
3rd configuration - u,x,u,u,x,u,x,u,x
4th configuration - u,x,u,x,u,u,x,u,x
5th configuration - u,x,u,x,u,x,u,u,x
6rd configuration - u,u,x,u,x,u,x,u,x
7th configuration - x,u,u,x,u,x,u,x,u
8th configuration - x,u,x,u,u,x,u,x,u
9th configuration - x,u,x,u,x,u,u,x,u
10th configuration - x,u,u,u,x,u,x,u,x
11th configuration - x,u,x,u,u,u,x,u,x
12th configuration - x,u,x,u,x,u,u,u,x
So, what we need to figure out after figuring out the above is the number of ways in which the labelled books can be ordered and the number of ways in which the unlabelled books can be ordered, multiply these values together and then multiply by 12.
I know it's highly probable that the answer obtained by solving the question this way will be wrong. I also know that there's a neater way of doing it. Can anyone show me the neat way?
Here's my sorry attempt at trying to solve the problem:
There are only 12 different configurations of 'slots' into which we can put the labelled books so that they are separated. This is shown below. The slots are represented by 'x' below. The unlabelled books are represented by 'u'.
1st configuration - x,u,x,u,x,u,x,u,u
2nd configuration - u,x,u,x,u,x,u,x,u
3rd configuration - u,x,u,u,x,u,x,u,x
4th configuration - u,x,u,x,u,u,x,u,x
5th configuration - u,x,u,x,u,x,u,u,x
6rd configuration - u,u,x,u,x,u,x,u,x
7th configuration - x,u,u,x,u,x,u,x,u
8th configuration - x,u,x,u,u,x,u,x,u
9th configuration - x,u,x,u,x,u,u,x,u
10th configuration - x,u,u,u,x,u,x,u,x
11th configuration - x,u,x,u,u,u,x,u,x
12th configuration - x,u,x,u,x,u,u,u,x
So, what we need to figure out after figuring out the above is the number of ways in which the labelled books can be ordered and the number of ways in which the unlabelled books can be ordered, multiply these values together and then multiply by 12.
I know it's highly probable that the answer obtained by solving the question this way will be wrong. I also know that there's a neater way of doing it. Can anyone show me the neat way?