2820643

6 exams in 18 days
https://math.stackexchange.com/questions/2820643/elementary-combinatorial-exam-allocation-problem

I have 18 days on which exams can take place, 6 exams to take, 2 exam slots per day (AM/PM). Only one exam is allowed per slot, allocation is random.

What is p:=P(2 days will have 2 exams on them)?

The species formula for one day is, meaning that there are three cases : two empty slots, one allocated-one empty, or two allocated slots.

$$Day(X) = \mathit 2X + t.X\cdot X + t^2.p.\mathit 2X$$

Here t is a counter for exams and p is a counter for double-exam days.

The session is the product of the 18 days.

$$ Session(X_1, X_2,...) = \prod_{i=1}^{18} Day(X_i) $$

The egf of the first day is

$$ e.g.f. = \frac {x_1^2} 2 + t.x_1.x_1+ t^2.p. \frac {x_1^2} 2 = (1+2t+p.t^2) \frac  {x_1^2} 2 $$

The coefficient of $$t^6.p^2$$ is $$ good = 73440 $$;

The class equation of the six exams session is

$$ 816*p^3 + 73440*p^2 + 685440*p + 1188096$$

For entire session,

The coefficient of $$t^6 $$ is $$ all = 1947792$$

$$ \frac {good}{all} = 0.0377$$