A physical neural network is to be built from 30 nodes split across 3 levels: input nodes, hidden nodes and output nodes. Every hidden node is connected directly to all the input nodes and also to all the output nodes. There are no direct connections between input nodes and output nodes, or between nodes within the same level.

(a) How many different configurations are there for a neural network with these specifications?

(b) How many of the neural networks in part (a) have a different number of nodes on each level?

(c) After considering the merits of the different options in part (a), the designers settle on having 5 input nodes, 15 hidden nodes and 10 output nodes. They then decide to run a preliminary test of part of the network. The test involves one input node, four hidden nodes and two output nodes. In how many ways can they choose the nodes that are involved in their test?

(d) Two separate tests are performed as described in part (c). For each test the nodes that are chosen on each level are selected uniformly at random from the nodes on that level. What is the probability that the set of nodes used for the second test does not include any of the nodes that were involved in the first test?