Question

There are six periods in each working day in a school in which $5$ subjects can be arranged if each subject is allotted at least one period and no period remains vacant is

Answer 1

Each of the arrangements which can be made by taking some or all of a number of things is called a permutation.

There are $5$ subjects,Hindi,English,Maths,Science and Social studies.

There are $6$ periods.

The first period can be allotted to any of the five subjects.

That is, there are $5$ different ways of allotting the first period.

Now, one subject has been allotted.

The second period can be allotted to any of the four subjects.

That is, there are $4$ different ways of allotting the second period.

Now, two subjects has been allotted.

The third period can be allotted to any of the three subjects.

That is, there are $3$ different ways of allotting the third period.

Now, three subjects has been allotted.

The fourth period can be allotted to any of the $2$ subjects.

That is, there are $2$ different ways of allotting the fourth period.

Now, four subjects has been allotted.

The fifth period can be allotted to any of the $1$ subject.

That is, there is only one way of allotting the fifth period.

Now, five subjects has been allotted.

Thus,by the principle of counting there are

$=5\times 4\times 3\times 2\times 1=120$ different ways to allot $5$ periods.

That is total number of arrangements

$=(6-1)(5\times 4\times 3\times 2\times 1)$

$=5\times 120=600$ arrangements