Permutations and school timetable If there are 6 periods in each working day of a school. In how many different ways can one arrange 5 subjects such that each subject is allowed at least one period? I tried this way- One of the six periods can be arranged in 5 ways and the remaining 5 periods in 5 factorial ways. Totally 600 ways

Lets think how we might pick this time table we can first assign a slot for each of our five subjects then finally pick the subject for the remaining slot.
Subject A has a choice of 6 slots
Subject B has a choice of 5 slots
Subject C has a choice of 4 slots
Subject D has a choice of 3 slots
Subject E has a choice of 2 slots
So far we have created $6! = 720 $ distinct permutations and there is a choice of 5 subjects for the remaining slot creating $5 \cdot 720 = 3600$ possible permutations but we note that these are not all distinct because we could swap the repeated subject with itself in the other slot the total number of distinct 
timetables is therefore $\dfrac{3600}{2} = 1800.$
