How many ways can 6 items be arranged in 4 boxes with restrictions?

[spaghetti3451]

Homework Statement

4 books by Shakespeare, 2 books are Dickens and 3 by Conrad are chosen for the problem. The question is to find the number of ways in which the books can be arranged s.t. the 3 Conrad books are separated.

Homework Equations

The Attempt at a Solution

n(C separated) = n(w/o any restrictions) - n(3 C's together) - n(2 C's together).

n(w/o any restrictions) = 9! because there are 9 items to be put in 9 places.

n(3 C's together) = 3! * 7! because the 3 C's form a cluster: 3! for items within the cluster and 7! for all the items, considering the cluster as an item.

n(2 C's together) = ... This is the tricky one as only 2 C's form a cluster and the number of places available for the other C depends on whether the cluster is at either of the edges of not.


The answer is supposed to be 151,200 (from the back of the textbook), but I can't happen to get it.

 

[tiny-tim]

hi failexam!

n(3 C's together) = 3! * 7! because the 3 C's form a cluster: 3! for items within the cluster and 7! for all the items, considering the cluster as an item.

good!

but probably easier to start again, this way …

the three conrad books have 4 "boxes" between and around them …

how many ways to fit 6 items into 4 boxes, with only the outside boxes allowed to be empty?

 

[rcgldr]

The three conrad books have 4 "boxes" between and around them … how many ways to fit 6 items into 4 boxes, with only the outside boxes allowed to be empty?

Note the middle 2 boxes have to have at least 1 book each, so that leaves the remaining 4 books to be placed in the 4 boxes in any pattern.