$10$
nearsighted people all put their glasses in a bin.

Andy, Brandy, Candy, and Dandy all have the same prescription.

Endy, Fandy, and Grandy have the same prescription, which is different from the other seven people.

Handy and Indy have the same prescription (again, which is different from the rest of the people).

Jandy's prescription is unique.

If the probability after they all randomly choose a pair from the bin that they can all see clearly (long distance) is $\frac{1}{b}$ , what is $b$ ?

The answer is 12600.

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The number of ways they can pick the glasses is $10!$

The number of ways they can all choose glasses with the right prescription is $4! \cdot 3! \cdot 2! \cdot 1!$

So, the probability is:

$P = \frac{4! \cdot 3! \cdot 2! \cdot 1!}{10!} = \frac{1}{\boxed{12600}}$