Maths
General
Easy
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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We know that the formula of dividing m different things into groups of sizes where
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But this is applicable on if the groups are of unequal sizes.
In the given problem all the groups are of size 3, and there are 5 groups.
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Again, the equal groups will be arranged amongst themselves, which is possible in 5! ways. Thus, we can say that the required number of ways in which the 15 different books can be divided into 5 groups of size 3 is
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5 heaps of 3 books each are to be made from 15 different books. We are to find in how many ways this can be done.
We know that the formula of dividing m different things into groups of sizes where
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Again, the equal groups will be arranged amongst themselves, which is possible in 5! ways. Thus, we can say that the required number of ways in which the 15 different books can be divided into 5 groups of size 3 is
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In the given problem all the groups are of size 3, and there are 5 groups.
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