Question

The number of ways of distributing 50 identical things among 8 persons in which such a way that three of them get 8 things each, two of them get 7 things each and remaining three get 4 things each is equal to
A.$\dfrac{{8!}}{{3!2!}}$
B.$\dfrac{{8!}}{{{{\left( {3!} \right)}^2}2!}}$
C.$\dfrac{{50!8!}}{{{{\left( {3!} \right)}^2}2!}}$
D.$\dfrac{{8!}}{{3!}}$

Answer 1

We will divide the things distributed to students given in the question as 888, 77, 444 since the things are identical and then, we will use the formula $n!$ to calculate the number of ways in which they can be distributed individually and then for distribution among 8 persons, we will calculate by dividing the total number of ways with the possible arrangements repeated over.

Complete step-by-step answer:
We are given that 50 identical items are required to be distributed among 8 persons. It is given that three out of them get 8 things, 2 get 7 and 3 get 4 things.
We are required to calculate the number of ways in which such a distribution can be done.
Now, for the distribution, the total number of ways in which 8 persons can get things in a particular order (using the formula: $n!$ where n is the total number of persons) is $8!$.
Now, we are told that three out of them get 8 things, 2 get 7 and 3 get 4 things.
We know that a person can get n things in $n!$ ways.
Therefore, three people can get 8 things in $3!$ ways and two people can get 7 things in $2!$ ways and three people can get 4 things for $3!$ ways. Now, we can see that $3!$ ways can be arranged individually in $2!$ ways.
Hence, this preparation can be done in $3!3!2!2!$ ways.
Now, the number of ways will be $8!2!$ ($\because$ since we can either give the group of three persons 8 things or 4 things as it is not specified).
Therefore, the total number of ways to distribute 50 things among 8 people in the manner that three out of them get 8 things, 2 get 7 and 3 get 4 things will be:
$\Rightarrow$ Total ways = $\dfrac{{8!2!}}{{3!3!2!2!}}$
Or, total ways = $\dfrac{{8!}}{{3!3!2!}} = \dfrac{{8!}}{{{{\left( {3!} \right)}^2}2!}}$
Hence, option (B) is correct.

Note: In this question, you may get confused in the steps where we have calculated the number of ways to do the individual arrangement of distribution of things among 8 persons. You may go wrong while calculating the number of ways by multiplying the total possible ways with $2!$ since there are two groups of 3 people and things can be divided in them in $2!$ ways.