Question

There are $7$ periods in each working day of a school. In how many ways can one organize $6$ subjects such that each subject is allowed atleast one period?

• A. $120$
• B. $126$
• C. $1440$
• D. $15120$

Answer 1

The correct option is D $15120$

There are $6$ subjects and $7$ periods. Each subject is allowed atleast one period.

Therefore, two periods will have the same subject and the remaining five periods will have different subjects.

Two periods can be selected from $7$ periods that have the same subject in ${}^7{C}_2$ ways.

Subject (same) allocated to these two periods can be selected in ${}^6{C}_1$ ways.

Now, we are left with five subjects and five periods. Remaining $5$ subjects can be arranged in the remaining $5$ periods in $5!$ ways.

Required number of ways
$={}^7{C}_2\times {}^6{C}_1\times 5!$
$=21\times 6\times 120$
$=15120$

Therefore, option (d.) is the correct answer.