Question

On each working day of a school there are six periods. The number of ways in which five subjects are arranged if each subject is allotted at least one period and no period remains vacant is

• A. 360
• B. 1800
• C. 120
• D. 210

Hint:

When allocating periods to subjects with restrictions, first determine the number of ways to allocate the extra period and then calculate the number of ways to assign the remaining periods.

Answer 1

The Correct Option is B

Step 1: Total periods and subjects.
There are 6 periods in total and 5 subjects to be allotted. Each subject should be allotted at least one period, and no period should remain vacant.

Step 2: Distribute the 6 periods.
We have to ensure that no period remains vacant. Since 5 subjects must be allotted 6 periods and each subject must get at least one period, one subject will be allotted 2 periods, while the other 4 subjects get 1 period each.

Step 3: Choose which subject will have 2 periods.
There are 5 subjects, so there are 5 choices for the subject that will be allotted 2 periods.

Step 4: Assign periods to the chosen subject.
For the subject that will be allotted 2 periods, we need to choose 2 periods out of the 6. The number of ways to choose 2 periods from 6 is given by:
\[ \binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15 \]

Step 5: Assign remaining periods to the other subjects.
Now, we need to allot the remaining 4 periods to the 4 remaining subjects. The number of ways to do this is \( 4! \), because each subject gets 1 period and there are 4 subjects.

Step 6: Calculate the total number of ways.
The total number of ways to assign the subjects to the periods is: \[ 5 \times \binom{6}{2} \times 4! = 5 \times 15 \times 24 = 1800 \]

Step 7: Conclusion.
Hence, the total number of ways to arrange the subjects is 1800. Therefore, the correct answer is:
\[ \boxed{1800} \]