Note: To understand the solution please go through the definitions of perfect matching
The complete graph $k_{n}$ have a perfect matching only when n is even. So let $n=2m.$
Let the vertices be $V_{1} , V_{2},\ldots ,V_{2m}$._{ }
$v_{1}$ can be joined to any other $\left(2m-1\right)$ vertices.
$v_{2}$ can be joined to any other $\left(2m-3\right)$ vertices.
Similarly, go till $V_{2m}$ which will have only one vertex to be joined with.
No. of Perfect matches$= (2m-1)(2m-3)(2m-5)\ldots(3)(1)$
In the above question $2m=6.$
So, No. of perfect matches = $5\times 3\times 1=15.$