2990035

https://math.stackexchange.com/questions/2990035/probability-with-sets

1) In Species language, the structure in the problem is *a 2-set of 5-sets of A or B or X

$$E_2(E_5(A+B+X))$$ having the e.g.f

$$ {1 \over 2!}(\frac {(a + b + x)^5} { 5!})^2$$

where X stands for the species of "1", Y for the species of "2" and Z for all other figures.

we are interested in the coefficient $$ {a\over 1!} {b \over 1! } {x^8\over 8!} $$ which is 126.

There are then 126 way of partitioning 10 elements among 2 of them are marked.

The good configurations are $$A.B.E_3(X).E_5(X)$$

The e.g.f is

$$a.b.{x^3 \over 3!}{x^8 \over 8!}$$ and the coefficient of

$$ {a \over 1!} {b \over 1! } {x^8\over 8!} $$ is 56.

Then we have to apply the definition of a probability.