2821613

[https://math.stackexchange.com/questions/2821613/understanding-solution-to-the-probability-problem-from-sheldon-ross-book A closet contains 10 pairs of shoes. If 8 shoes are randomly selected (Ross) ]

A closet contains 10 pairs of shoes. If 8 shoes are randomly selected, what is the probability that there will be

- no complete pair;

- exactly one complete pair

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

Here t is a counter for the mumber of shoes present in selection and p is a counter for a complete pair.

The closet is the product of the 10 days.

$$ Closet (X_1, X_2,...) = \prod_{i=1}^{10} Day(X_i) $$

we are interested in the coefficient of $$ t^8.\prod_{i=1}^{10} {x_i^2 \over 2!} $$

$$210.d^4 + 10080.d^3 + 50400.d^2 + 53760.d + 11520$$

no complete pair
The type 8 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 type is described by :

exactly one pair
For the type 8 = 2 + 1 + 1 + 1 + 1 + 1 + 1 we have :

$$ X \cdot E_2(Y_X) \cdot E_6(X \cdot Y_X \cdot Y_X) \cdot E_3 (X \cdot E_2(Y_X))$$

The egf is

$$    \frac 1 {1!}  \left( x \frac {y^2} {2!} \right ) . \frac 1 {6!} \left( x.y.y \right )^6 . \frac 1 {3!} \left( x \frac {y^2} {2!} \right )^3 = 53760 \frac 1 {10!} \left( x \frac {y^2} {2!} \right )^{10} $$