Counting Couples A group of m men and w women randomly sit in a single row at a theater. If a man and woman are seated next to each other, they form a "couple." "Couples" can overlap, which means one person can be a member of two "couples." 
Question: What is the expected number of couples?
Comment:
I have a hard time with word problems that deal with "expectations". 

Continuing the computation we can calculate $E[C(C-1)]$.
We have $$\left.\left(\left(\frac{d}{dz}\right)^2 (P+Q)\right)\right|_{z=1} =
2\,{\frac {uv \left( -2\,uv-u+{u}^{2}+{v}^{2}-v \right) }{ \left( v-1+u \right) ^{3}}}.$$
After a straightforward calculation this transforms into
$$ E[C(C-1)] = 
{\frac {2\,mw \,\left( 2\,mw-w-m \right) }{ \left( m+w-1 \right)  \left( m+w \right) }}.$$
