The total population size is N = 5, of which some are diseased and the rest are healthy. During any single period of time, two people are selected at random from the population and assumed to interact. The selection is such that an encounter between any pair of individuals is independent of any other pair. If one of the persons is diseased and the other isn't, the with probability 0.1 the disease is transmitted to the healthy person. Otherwise, no disease is transmitted. Let $\displaystyle X_n$ denote the number of diseased people in the population at the end of the nth period. Find the transition probability matrix.

I did the TPM for two people from the population:

H ~ Healthy Person D ~ Diseased person

$\displaystyle \ H \ \ \ \ \ D$

$\displaystyle \begin{bmatrix}0.9 & 0.1\\ 0 & 1\end{bmatrix}$

But I'm not sure how to account for the N=5 population