Question

In how many ways 5 different books can be distributed among Ram, Shyam and Gopal such that two persons get 2 and one gets 1 book?

Answer 1

Hint: We will calculate the number of ways to distribute the 5 different books into 3 stacks such that no stack has more than 2 books. Then we will find the number of ways in which the three stacks of books can be distributed among Ram, Shyam, and Gopal. We will use the combinations notation and definition to formulate the expression for the counting number of ways.

Complete step by step answer:
We have 5 different books. We will split them into 3 stacks such that no stack has more than 2 books. So, in the first stack, we can choose any two books out of the five, which is ${}^{5}{{C}_{2}}$. Then for the second stack, we are left with three books. We will choose 2 out of the three books, which is ${}^{3}{{C}_{2}}$. For the last stack, we will be left with only one book. So there is only one choice. Now, the total number of ways in which these stacks will be filled by books is as follows,
$\text{number of ways the 5 books are stacked = }{}^{5}{{C}_{2}}\times {}^{3}{{C}_{2}}\times 1$.
We know that ${}^{n}{{C}_{r}}=\dfrac{n!}{r!\left( n-r \right)!}$. Using this definition, we get
\[\begin{align}
  & \text{number of ways the 5 books are stacked = }\dfrac{5!}{2!\cdot 3!}\times \dfrac{3!}{2!\cdot 1!}\times 1 \\
 & =\dfrac{5\times 4}{2}\times 3 \\
 & =30
\end{align}\]
So, the 5 different books can be stacked in 30 ways. Now, these stacks can be assigned to Ram, Shyam and Gopal in 3 ways.
Therefore, the number of ways 5 different books can be distributed among Ram, Shyam and Gopal such that two persons get 2 and one gets 1 book is $30\times 3=90$.

Note:
The same approach can be used for different numbers of books or different numbers of people. It is necessary to understand how the combinations, ${}^{n}{{C}_{r}}$ works. There is a possibility to mess up the calculations. So it is important that we explicitly write all cases that are considered for better clarity. The factorial of a number is the product of all the numbers before it with the number itself.