From the text: If 3-digit numbers are selected so that they differ from each other in two or more positions, then the total possible number selected is not unique but ranges from 50 to 100. This article examines some problems in the random selection of 3-digit numbers; the study originated in the following manner. In a regional mathematics contest, students were to be examined so that their names remained anonymous. Each student was assigned a unique random 3-digit identification number to be written, in place of his or her name, on the answer sheet. To reduce the possibility of a mix-up associated with any student putting down the incorrect identification number, the numbers were randomly selected so that each identification number differed from every other one in at least two positions. A computer program was written that randomly selected 3-digit numbers, each of which differed from the others in at least two positions. The computations in the program continued until no more numbers could be selected satisfying the required condition. However, each time the program was run, a different number of identification numbers was generated. This surprising result motivated the present study. Of particular interest was the minimum number, the average number, and the maximum number of identification numbers that could be generated. After some notation is defined, an analysis of the problem is presented. Additional research, suggested by this study, is accessible to undergraduate students.