# Probability of inter-group links in a network with maximum degree 1

In an undirected network, there are two groups of nodes. Group 1 has N1 nodes, and group 2 has N2 nodes.

The links in the network are generated following such rules:

(1) The maximum degree is 1, which means each node has at most 1 link.

(2) There should be as many links as possible in the network. In another word, if (N1+N2) is even, each node should have a neighbor. If (N1+N2) is odd, only one node can be not connected with other nodes.

(3) Each node has the same probability of linking any other nodes.

"Inter-group link" is defined as the link connecting two nodes in two different groups.

[Question] What is probability that there are K inter-group links in this network?