Samir had prepared the problem tests for Stages 1 to 5 of Geometry and Combinatorics for next week but forgot to label which test was for which stage. Since Samir didn't label them, the computer assigned them labels 1 through 5 randomly, with each label appearing only once. The probability that the problems given to each stage are within one stage of what they were supposed to be can be expressed as a/b, where a and b are positive, coprime numbers. What is the value of a+b?
Details and assumptions
The computer randomly assigns each problem test to a stage, and each stage has exactly 1 problem test that is assigned. For example, the computer could assign the stage 1 problem test to stage 5 students, the stage 2 problem test to stage 4 students, the stage 3 problem test to stage 3 students, the stage 4 problem test to stage 2 students and the stage 5 problem test to stage 1 students.