Jóska and Gábor every summer watched the license plates of 51 cars passing by each day, and if all four digits of the number were different, Jóska got 1 point, otherwise Gábor did. After 40 days, Gábor claimed that he won because he had more points on 21 days. However, Jóska claims that if they added up the results of all 40 days, he would have more points. - In a long game, who has a better chance of accumulating more points? Based on what assumptions did we answer the question?
An example of the current Hungarian car license plate system is: AG-0246. We assume the following: for every pair of letters (e.g., AG), every number in the series of ten thousand (0000-9999) has been issued; the boys could count every license plate during the long game (including those no longer in circulation), and each one only once.

Thus, it is sufficient to compare how many license plates in a single series of ten thousand are favorable to József and how many to Gábor. Considering József's numbers, for the first position, we can choose any of the 10 digits, for the subsequent positions, 9, 8, and 7 respectively, as, for example, the digit in the first position cannot be repeated in the second. According to this, 10 * 9 * 8 * 7 = 5040 license plates are favorable to József, and 4960 are favorable to Gábor, as in the latter, not all digits differ from the ones preceding them. Thus, under our assumptions, József has a better chance.

Of course, we know that our simplifying assumptions do not fully reflect the complexity of real life. The boys' roughly 2000 observed license plates are, in any case, a rather small sample compared to the possibilities. With different assumptions, we might have arrived at a different result. For example, if only the first 123 numbers of a series were issued - including 0000 - this would all be favorable to Gábor. (This is a common observation with special-purpose vehicles.) It is also common to see license plates of the type 99xy, as if perhaps the plates are issued in reverse order; this is also favorable to Gábor.

Dombi Gábor (Szeged, Ságvári E. Gymn., II. grade) Mester Gábor (Szombathely, Nagy Lajos Gymn., II. grade) Máté András (Budapest, Kölcsey F. Gymn. I. grade)