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?
Solution:
To find: number of ways of arranging 5 subjects in 6 periods.
Condition: at least 1 period for each subject.
5 subjects in 6 periods can be arranged in $P(6,5)$.
Remaining 1 period can be arranged in $\mathrm{P}(5,1)$
Formula:
Number of permutations of $n$ distinct objects among $r$ different places, where repetition is not allowed, is
$P(n, r)=n ! /(n-r) !$
Total arrangements $=\mathrm{P}(6,5) \times \mathrm{P}(5,1)=\frac{6 !}{(6-5) !} \times \frac{5 !}{(5-1) !}$
$=\frac{6 !}{1 !} \times \frac{5 !}{4 !}=720 \times 5=3600$
Total number of ways is 3600 ways.