Determine all possible pairs (x, y) of nonnegative integers that satisfy this equation:

x!=120*(y!)

Prove that there are no further pairs in consonance with the given conditions.

This curious fact actually ocurred to me yesterday. It never happened before.

To grant access to the operations provided by the ATM (Automatic Teller Machine) of my bank, I have to type (touching the screen) the password of my magnetic card.

The screen which appears to me shows five "buttons", each one of them labeled with two digits, from 0 to 9. For example, in the first there are the numbers 0-2, in the second, 3-7, in the third, 4-5, in the fourth, 1-9 and in the fifth, 6-8. So, if my password, that consists of 6 digits, not necessarily different, is 123456, I touch, in order, the fourth button (1), the first (2), the second (3), the third (4), again the third (5) and the fifth (6).

The numbers that appear in each button change daily, and yesterday I noticed that to enter my password, I touched only two buttons. To clarify, with the configuration above, if my password were "357457", I would have touched the second button, the third, the second, the third, again the third, and the second.

What is the probability that this occurs, that is, that I have to touch only (and exactly) two buttons to enter my password, if it is made of a) 2 different digits; b) 3 different digits; c) 4 different digits?

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