Question

There are 5 different Jeffrey Archer books, 3 different Sidney Sheldon books and 6 different John Gresham books. Find the number of ways in which at least one book can be given away.

Answer 1

Hint: First, list the possible options for each book to be decided. Then find the possible option for Jeffrey Archer books, Sidney Sheldon books and John Gresham books. After that multiply these 3 options to find the total ways in which it can be decided. There will be only one way in which no book would be given. So, subtract it from total possible ways. Thus, the remaining ways are the desired result.

Complete step-by-step answer:
Given: - There are 5 different Jeffrey Archer books, 3 different Sidney Sheldon books and 6 different John Gresham books.
As every book has 2 options either give or don’t give.
Now, there are 5 different Jeffrey Archer books. So, the total number of options will be,
${2^5} = 32$
Also, there are 3 different Sidney Sheldon books. So, the total number of options will be,
${2^3} = 8$
Also, there are 6 different John Gresham books. So, the total number of options will be,
${2^6} = 64$
Total ways will be,
\[32 \times 8 \times 64 = 16384\]
There will be only one way in which no book would be given.
So, the number of ways in which at least on book can be given away is,
$16384 - 1 = 16383$

Hence, the number of ways in which at least one book can be given away is 16383.

Note: The fundamental principle of counting is used to find the total ways.
The fundamental counting principle is a rule used to count the total number of possible outcomes in a situation. It states that if there are $n$ ways of doing something, and $m$ ways of doing another thing after that, then there are $n \times m$ ways to perform both of these actions. In other words, when choosing an option for $n$ and an option for $m$, there are $n \times m$ different ways to do both actions.