There was a problem I came across which left me in thinking for a whole day and I wanted to share the generalised solution or theory with you

Jan and 5 other students are in classroom and 2 of the students will be selected to clean the blackboard. What is the probability to
a): That Jan will be among the 2 selected students to clean the blackboard today
b) That Jan will be among the 2 selected students to clean the blackboard today and tomorrow such that 1 student will clean the blackboard today and the other student will clean the blackboard tomorrow. I can also be the case that the same student will clean the blackboard both the days.


Solution:

In this we will be tempted to use the same theory in both the parts that we will use combinations and then will calculate probability.

BUT

when we will select the students for different days, then the order will be pertinent and hence permutations is also use. I will explain

A) If Jan is selected among the 2 students, then 5C1 are the ways in which 2nd student will be selected and total ways will be = 6C2
probability = 5C1/6C2


B) In this Jan can do the work today or tomorrow or both, so

Today = 1/6 * 5/6
Tomorrow = 5/6 * 1/6
Both days = 1/6 * 1/6

Total probability = 1/6 * 5/6 + 11/6 * 5/6 + 1/6 * 1/6