Question

In how many ways can 3 books be given to 5 persons if,

1. No person can have more than one book.
2. A person can have any number of books.

Answer 1

Here, we have to find the no. of ways in which the 3 books to be given to 5 persons. In the first condition no person can receive more than three books so in this question we will apply permutation and in the second question a person can receive any number of the books so in question we will apply both permutation and combination both.

Complete step by step solution:
Here we have given that we have to distribute 3 books to 5 different persons
No person can have more than 1 book.
We have to give 3 books to 5 people.
Now, consider 5 persons are standing in a one line
The three books will be going to only three people out of five.
And number of ways choosing 3 out of 5
$= {}^5{P_3} \\\ = \dfrac{{5!}}{{(5 - 3)!}} \\\ = \dfrac{{120}}{2} =60$
Therefore, number of ways of choosing 3 out of $5 = 60$ ways
A person can have any number of books.
Case 1: No person has more than one book
The case one is same the condition we use above then there are 60 ways
Case 2: one person receiving 2 books,
As all the books are different the number of ways of selecting two books out of three $= {}^3{C_2} \Rightarrow 3 \text{ Ways}$
So, there are 5 ways to distribute 2 books to any one person out of five.
And the remaining one book can distribute to four persons in four different ways
$\therefore$ Total no. of ways $= 3 \times 4 \times 5$
=60 Ways
Case 3: All the three books to one person,
We can do that in 5 ways because we can give all of them to any of them
$\therefore$ Total number of ways of distributing 3 different books to 5 people such that each one can receive any number of books
$\Rightarrow 60+60+5 \\\ \Rightarrow 125 \\\ \Rightarrow 125 \text{ Ways}$

Note:
Here, the permutation is used when the order is important and the combination is used when the order is not important. Here, we will multiply the number of ways when they are depending on each other and add if they are independent of each other.