If there are 6 periods in 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?

while solving this problem why we write ^{6}P_{5} ??????

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 5 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 4 subjects.

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

Now two subjects have been allotted.

The third period can be allotted to any of the 3 subjects.

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

Now three subjects have 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 have been allotted.

The fifth period can be allotted to 1 subject.

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

Now five subjects have been allotted.

Thus, by the principle of counting there are different ways to allot 5 periods.

There are 6 periods in total.

Hence the 6th period can be allotted to any of the 5 subjects.

Thus, the total number of arrangements of 5 subjects in 6 periods is

That is total number of arrangements is

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