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We prove that the number of perfect matchings in ${\mathcal G}(n,d)$ isasymptotically normal when $n$ is even, $d\to\infty$ as $n\to\infty$, and$d=O(n^{1/7}/\log^2 n)$. This is the first distributional result of spanningsubgraphs of ${\mathcal G}(n,d)$ when $d\to\infty$.  Moreover, we prove that ${\mathcal G}(n,d-1)$ and ${\mathcal G}(n,d)$ can becoupled so that ${\mathcal G}(n,d-1)$ is a subgraph of ${\mathcal G}(n,d)$ withhigh probability when $d\to\infty$ and $d=o(n^{1/3})$. Further, if$d=\Omega(\log^7 n)$, $d=O(n^{1/7}/\log^2n)$, and $d\le d'\le n-1$ then${\mathcal G}(n,d)$ and ${\mathcal G}(n,d')$ can be coupled so thatasymptotically almost surely ${\mathcal G}(n,d)$ is a subgraph of ${\mathcalG}(n,d')$.

The number of perfect matchings, and the nesting properties, of random regular graphs