Question

Out of 6 books, in how many ways can a set of one or more books be chosen?
A. \[64\]
B. \[63\]
C. \[62\]
D. \[61\]

Answer 1

Hint: First, calculate the number of ways of choosing no book from 6 books. Then, find the number of ways all books can be chosen. In the end, subtract the number of ways of choosing no book from 6 books from the number of ways all books can be chosen and solve it to get the required answer.



Formula Used:The combination formula: \[{}^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\]
\[0! = 1\]



Complete step by step solution:Given:
Total number of books: 6
The total number of ways of choosing no book from 6 books: \[{}^6{C_0}\]
Solve it by applying the combination formula.
\[{}^6{C_0} = \dfrac{{6!}}{{0!\left( {6 - 0} \right)!}}\]
\[ \Rightarrow {}^6{C_0} = \dfrac{{6!}}{{1 \times 6!}}\]
\[ \Rightarrow {}^6{C_0} = 1\] \[.....\left( 1 \right)\]

Now find the number of ways all books can be chosen.
Since there are 6 books.
So, the number of ways all books can be chosen: \[{2^6}\] \[.....\left( 2 \right)\]

To calculate the number of ways can a set of one or more books be chosen subtract equation \[\left( 1 \right)\] from the equation \[\left( 2 \right)\].
We get,
\[{2^6} - {}^6{C_0} = 64 - 1\]
\[ \Rightarrow {2^6} - {}^6{C_0} = 63\]



Option ‘B’ is correct



Note: The number of ways all books can be chosen is also calculated by adding the number of ways of choosing one book, the number of ways of choosing two books, and so on up to the number of ways of choosing all 6 books.
i.e., \[{}^6{C_1} + {}^6{C_2} + {}^6{C_3} + {}^6{C_4} + {}^6{C_5} + {}^6{C_6}\]