find the number of ways of arranging 4 subjects in 5 periods such that all the subjects will be there in each arrangement...

We know that all the subjects have to be there. So first of all, the data that we have is, we have four periods, each of which have to come atleast once in the arrangement. Now, taking on the remainig period (the fifth one).

Any one of these four peroids can be the fifth one without any restriction, but it will certainly be a repitition.

Therefore, first of all, we select one amongst the four periods, by making a selection.

C(4,1) {if that means what we pronounce as 4C1}.

Now, apparently we have five subjects which are to be arranged in five periods with a single repetition. So we manage it to be 

[C(4,1).P(5,5)]/(2!) 

which is the required answer.

CHEGCK

There are 96 ways of arranging 4 sub. in 5 periods.