Question

There are 6 periods in each working day of a school. In how many ways can one organize 5 subjects such that each subject is allowed at least one period?
A. 3200.
B. 1800.
C. 3600.
D. none of these.

Answer 1

Hint: To solve this question, we will use the concept of permutation and combination. The number of combinations of n different things taken r at a time, denoted by ${}^n{C_r}$, is given by ${}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}},0 \leqslant r \leqslant n.$ and the number of permutations of n objects, where p objects are of the same kind and rest all are different = $\dfrac{{n!}}{{p!}}$.

Complete step by step solution:
Given that,
Total number of subjects = 5.
Total periods in each working day = 6.
We can see that 5 subjects can be studied in the 5 out of 6 periods and one subject will get repeated.
So, the number of ways to select the repeating subject for that 6th period = ${}^5{C_1} = 5$ways.
Now, the 5 subjects that are allowed in 6 periods can be arranged in different ways.
The number of different arrangements of 5 subjects in 6 periods are = $\dfrac{{6!}}{{2!}} = \dfrac{{6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1}} = 360.$
Hence, the number of ways to organize 5 subjects in 6 periods such that each subject is allowed at least one period = ${}^5{C_1} \times \dfrac{{6!}}{{2!}} = 5 \times 360 = 1800$ ways.
Therefore, the correct answer is option (B).

Note: In this type of questions, we have to remember some basic points of permutation and combination. First, we will arrange the 5 subjects simply in 5 periods and then we will find out the number of ways to arrange the 5 subjects in that one 6th period. After that we will multiply the number of ways to select 5 subjects in one period with the number of ways to arrange the 5 subjects in 5 of 6 periods. Through this, we will get the required answer.