Expectation values of male-male, male-female, female-female pairs N people are seated at random to form a circle, among whom N1 are male. What are the expected numbers of male-male, male-female, and female-female nearest-neighbor pairs?

Label the chairs $1$ to $N$, counterclockwise, and assume $N\gt 1$.  Let $X_i=1$ if the person in Chair $i$ is male and his neighbour in the counterclockwise direction is male. Let $X_i=0$ otherwise. Then the number $X$ of male-male nearest neighbour (unordered) pairs is given by
$$X=X_1+X_2+\cdots+X_N.$$
The $E(X_i)$ are all the same. So by the linearity of expectation we have $E(X)=NE(X_1)$. 
The probability that $X_1=1$ (and hence the expectation of $X_1$) is the probability the people in chairs $1$ and $2$ are both male. This probability is $\frac{N_1}{N}\cdot\frac{N_1-1}{N-1}$.
It follows that $E(X)=\frac{N_1(N_1-1)}{N-1}$.
The calculations for the other two questions are similar. 
Remark: Alternately, one can do a Total Expectation calculation. I prefer the indicator random variable version. 
