 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, givingThere 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 $\left(\genfrac{}{}{0ex}{}{1}{1}\right)\left(\genfrac{}{}{0ex}{}{7}{4}\right)\left(\genfrac{}{}{0ex}{}{8}{3}\right)=1960$.