Ok, so let me venture a possible solution.

The total possible combinations in set-1 locks is

C1(total) = 5x4x3x6 = 360

For set-2 we have:

C2(total) = 4x3x6x2 = 144

digit-1 has 2 common numbers between the respective pools of set-1 and set-2

digit-2 has 1 common numbers between the respective pools of set-1 and set-2

digit-3 has 3 common numbers between the respective pools of set-1 and set-2

digit-4 has 2 common numbers between the respective pools of set-1 and set-2

So there are C(common)=2x1x3x2=12 possible combinations that are common between the two sets of locks.

The chance that a potential common combination is produced by any set-1 lock is:

P1(common) = 12/360 = 0.033

The same chance for a common combination from any lock in set-2 is:

P2(common) = 12/144 = 0.083

Then I used an online calculator of binomial probability, once for each set of locks.

For set-1, I used:

probability of success P = 0.033

number of trials N = 300

number of successes K = 0

==> Binomial Probability that one of the 12 common combination never occurs after 300 trials P1(x=0) = 4.24e^-5

Then, for set-2:

probability of success P = 0.083

number of trials N = 20

number of successes K = 0

==> Binomial Probability that one of the 12 common combination never occurs after 20 trials P2(x=0) = 0.175

We want the probability that no common combination appears between the 300 set-1 locks and the 20 set-2 locks.

So we want that no potentially common combinations appear in set-1, or that no potentially common combinations appear in set-1, or that none of set-1 and set-2 produce a potentially common combination.

P(cumulative)= P1(x=0)+P2(x=0)+(P1(x=0)*P2(x=0)) = 0.000042 + 0.175 + (0.000042 * 0.175) =~ 0.175 or 17.5%

Can anyone please confirm that this solution is correct?

Thanks-