If there are 6 periods in each working day of a school, in how many ways can one arrange 5 - Maths - Permutations and Combinations

Question

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
6P5 ??????

Vimala · Accepted Answer

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 $5 \times 4 \times 3 \times 2 \times 1$ 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

$5 (5 \times 4 \times 3 \times 2 \times 1)$

That is total number of arrangements is

$(6 - 1) (5 \times 4 \times 3 \times 2 \times 1) = 6 (5 \times 4 \times 3 \times 2 \times 1) - 1 (5 \times 4 \times 3 \times 2 \times 1) = 6 \times 5 \times 4 \times 3 \times 2 \times 1 - 5 \times 4 \times 3 \times 2 \times 1 = {}^{6}{P_{5} - 5!}$

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 6P5 ??????

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 ⁶P₅ ??????