I will be having 18 Students in my class this year. I'd like to have them learn in pairs rotating every day with a different student in the class. What are all the possible pairings.
for example on Day 1: Student 1 learns with 2/ 3 with 4/5 with 6 etc etc
I need an 18 day rotation for this please
Thanks
This is equivalent to asking in how many ways a $2n-$element set can be partitioned into two-element subsets. Let $T_{2n}$ be that number.
Let's define
$T_0 = 1$
$T_2$ clearly equals $1$.
We can express $T_{2(n+1)}$ in terms of $T_{2n}$ and $T_{2(n-1)}$?
Let's consider the set with $2n+2$ elements $\{1,2,3,\dots,2n\}\cup\{a,b\}.$
If we choose $\{a,b\}$ to be in the partition, then we know that there are $T_{2n}$ ways to partition the remaining $2n$ elements.
We can choose partition elements of the form $\{a,u\}$ and $\{b,v\}$ where $1 \le u \ne v \le 2n$ in $2n(2n-1)$ ways and there will always be $T_{2n-2}$ ways to partition the remaining $2n-2$ elements.
This implies that $$T_{2(n+1)} = T_{2n} + 2n(2n-1)T_{2n-2}$$
The OEIS points out that
$$T_{2n} = (2n-1)!! = (2n-1)(2n-3)\cdots1$$
In either case
$T_{18} = 34459425$
