Question
The number of ways in which mn students can be distributed equally among m sections is
 (mn)^{n}



Hint:
In order to solve this question, we should have some knowledge regarding the concept of combination, that is for choosing r out of n items irrespective of their order we apply the formula, . And therefore, we will choose n number of students for each section turn by turn.
The correct answer is:
Detailed Solution
In this question, we have been asked to find the number of ways in which we can distribute mn students equally among m section. Now, we have been given that there are mn students in total and they have to be distributed equally among m sections.
For the second section, we will again choose n students but out of (mn – n) because n students have already been chosen for the first section. So, we get the number of ways of choosing n students for the second section as
Similarly, for the third section, we have to choose n students out of (mn – 2n). So, we get the number of ways of choosing n students for the third section as
And we will continue it in the same manner up to all mn students will not be divided into m section.
So, for (m – 1)th section, we will choose n students from (mn – ( m – 2)n) student. So, we get the number of ways of choosing n students for (n – 1)th section, we get,
And for the mth section, we get the number of ways for choosing students as,
Hence, we can write the total number of ways of distributing mn students in m section as
And we can further write it as,
While solving this question, the possible mistake one can make is by always choosing n students for all sections from mn students which is totally wrong because at a time one student can only be in 1 section. So, if n students are selected for 1 section then in the second section, we will choose from (mn – n).
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