When admitting students to the university, each applicant is assigned a cover code consisting of five digits. The thorough, yet superstitious associate professor decided to eliminate from all possible codes (i.e., 00000 to 99999) those that contained the number 13, meaning the digit 3 immediately following the digit 1. How many codes did the professor have to eliminate?
First, we will determine how many codes contain the number 13 once:

Codes of type 13***: On the third, fourth, and fifth positions, any of the ten digits can appear, so there are $10 \cdot 10 \cdot 10=1000$ codes.

Codes of type *13**: Similarly, we find that there are 1000 codes.

Codes of type **13*: 1000 codes.

Codes of type ***13: 1000 codes.

In total, 4000 codes. However, these also include codes that contain two instances of the number 13. Now we will determine how many such codes there are:

Codes of type 1313*: 10 codes,

Codes of type 13*13: 10 codes,

Codes of type *1313: 10 codes.

In total, 30 codes with two instances of the number 13. These codes were counted twice before. Since we have exhausted all possibilities, the total number of codes is $4000-30=3970$.

The superstitious professor has eliminated 3970 codes.