There are 6 periods in each working day of a school. In how many ways can one organize 5 subjects such that each subject is allowed at least one period?

Answer 1

There are 5! = 120 ways to arrange 5 subjects.

Now, the 6th subject (which just happens to be one of the 5 as well) can be inserted in any place in the schedule. So, there are 6 places to insert it. The 6th subject can be any of the 5, so there are 5 ways to choose it.

So, there are 5! * 6 * 5 ways to arrange the subjects, but some arrangements will have the same subject for two periods in a row.