Question

If there are 6 periods on each working day of a school,in how many ways can one arrange 5 subjects such that each subject is allowed at least one period?

Answer 1

Total 1800 ways.

Explanation:

Under the conditions you mentioned in the question, one subject is assigned two periods and the other four subjects are assigned one period each.

Let the five subjects `A,B,C,D,E` and six periods `1` to `6`.

[Step1] Select the subject assigned two periods and then select two periods which it is assigned.

Number of selection for one subject is `""_5C_1=5` and selection for two periods is `""_6C_2=(6*5)/(2*1)=15`.
The total number of selection in Step1 is `5*15=75`.

[Step2] Assign the other four subjects to four periods.

This step is easy. The total number of selection in Step2 is
`""_4P_4="4!"=24`.

Therefore, there is `75*24=1800` total selections meeting the conditions.