Let
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
be a set of
![7](/media/m/5/1/9/519154d5119d15088eebb25b656d58dd.png)
different prime numbers and
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
a set of
![28](/media/m/8/a/2/8a23d80865b1d7ff9b4e2d8e34febaee.png)
different composite numbers each of which is a product of two (not necessarily different) numbers from
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
. The set
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
is divided into
![7](/media/m/5/1/9/519154d5119d15088eebb25b656d58dd.png)
disjoint four-element subsets such that each of the numbers in one set has a common prime divisor with at least two other numbers in that set. How many such partitions of
![C](/media/m/5/a/b/5ab88f3f735b691e133767fe7ea0483c.png)
are there ?
%V0
Let $P$ be a set of $7$ different prime numbers and $C$ a set of $28$ different composite numbers each of which is a product of two (not necessarily different) numbers from $P$. The set $C$ is divided into $7$ disjoint four-element subsets such that each of the numbers in one set has a common prime divisor with at least two other numbers in that set. How many such partitions of $C$ are there ?