A problem for the Sorting Hat

My secondary school, CRGS, admits (or used to admit–I’m not sure now) 100 people each year (technically, 96+4). They are to be split into h = 4 houses, such that the houses have equal numbers and siblings are in the same house as each other. What happens if they have a year of fifty pairs of twins, or twenty-five sets of quadruplets?

(I believe there’s also a condition on how the four houses should be distributed evenly across the forms, but for simplicity let us ignore it.)

More serious question: Let us call an intake unresolvable if the two conditions cannot be satisfied. For a given probability distribution of twins, triplets, etc., consider the probability P(n) that an intake of hn people will be unresolvable. What values of n are local minima of P(n)?

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