Question

There are six periods in each working day of a school. The number of ways in which 5 subjects can be arranged if each subject is allotted at least one period and no period remains vacant is:
A. 210
B. 1800
C. 360
D. 3600

Answer 1

Hint: First of all find out the total number of ways in which 5 subjects can be arranged in 5 periods. Then, find the possible ways for the sixth period when any of the five subjects can be selected. Finally, divide by 2! as there will be two same subjects for each arrangement.

Complete step-by-step answer:

We are given that there are six periods and five subjects are to be arranged in those six periods.
As the order of the period does not matter, we will use the combination to find out the number of arrangements possible.
If each subject is studied each day, then 5 subjects can arrange in 5 periods in \[^5{C_5}\]
Now, for the remaining period, any of the subjects can be selected.
Thus, from the given 5 subjects, 1 subject for the remaining period can be selected in the \[^5{C_1}\]
Similarly, there are 6 such subjects and can arrange themselves in 6! ways.
Also, while arranging the subjects, there will be two same subjects in each arrangement. Hence, we will divide by 2!
Therefore, we have \[\dfrac{{^5{C_5}{ \times ^5}{C_1} \times 6!}}{{2!}}\] numbers to arrange periods.
Solving the expression, we get,
\[\dfrac{{1 \times 5 \times 6!}}{{2!}} = \dfrac{{5 \times 6.5.4.3.2!}}{{2!}} = 1800\]
Thus, the total number of arrangements is 1800.
Hence, option B is correct.

Note: We should have knowledge about the combinations to solve the given question. As the order of the period does not matter, we will use combinations to find out the number of arrangements possible. The combination is represented as, \[^n{C_r}\], where \[n\] is the total number of items and \[r\] is the number of items selected. Also, \[^n{C_n} = 1\] and \[^n{C_1} = n\]. Also, the common mistake in this question is the students will not divide the equation by 2!, as we are given 2 subjects will be alike in each arrangement.