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A group of 6 men and 6 women is randomly divided into 2 groups of size 6 each (Ross)

A group of 6 men and 6 women is randomly divided into 2 groups of size 6 each. What is the probability that both groups will have the same number of men?

$$ \mathit6(X+Y) = \mathit6(X) + \mathit5(X) \cdot Y ... + t. \mathit3(X)\cdot \mathit3(Y) + ... + \mathit6(Y) $$

Unlabeled groups
$$ \mathit2(\mathit6(X+Y) \rightarrow {1 \over 2!} \left  (  { x^6 \over 6! } + { x^5y \over 5!1! } +...+ t{x^3y^3 \over 3!3!} + ...{ y^6 \over 6! } \right ) ^2 $$

$$ coeff(x^6y^6) = 200t^2 + 262 $$

the types equation is $$ 200+262 = 462 $$

Labeled groups
$$ \mathit6(X+Y) \cdot \mathit6(X+Y) \rightarrow  \left  (  { x^6 \over 6! } + { x^5y \over 5!1! } +...+ t{x^3y^3 \over 3!3!} + ...{ y^6 \over 6! } \right ) ^2  $$

$$ coeff(x^6y^6) = 400t^2 + 524 $$

the types equation is $$ 400+524 = 924 $$