= Solution
Each relevant cousin marriage is a cross between two <genetic carriers> of $q$. A child receives two copies of $q$ with <probability> $(1/2)(1/2)=1/4$, and fails to receive two with <probability> $3/4$. Conditional on the parental <genotypes>, distinct offspring transmissions are independent. There are eight specified double-copy children and eight specified remaining children, giving
$$
\boxed{P=(1/4)^8(3/4)^8=\frac{3^8}{4^{16}}.}
$$
There are no <binomial coefficients> because the subjects have been individually selected. If only the counts in the three sibships were specified, rather than their identities, the <probability> would instead be multiplied by $\binom11\binom74\binom83=1960$.
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