A student is allowed to select at most books from a collection of books. If the number of ways in which he can do this, is , then the value of is equal to
Explanation for the correct option :
Step-1 : (Find the total number of ways of choosing at most books from a collection of books)
Formula to be used : We know that the number ways of choosing objects from a collection of objects is .
Given that, a student is allowed to select at most books from a collection of books. So, he can choose book or book or books or …. or at most books.
Then, the number of ways he can choose book is .
The number of ways in which he can choose book is .
The number of ways in which he can choose books is .
The number of ways in which he can choose books is .
So, the total number of ways in which he can choose at most books from books .
Step-2 : Evaluate the expression .
Formula to be used : We know that
and
.
So, using the above formula, we get the following identities :
Then using the above identities, we obtain
Again, using the formula, we also have,
Thus we must get the following identity :
Step-3 : (Evaluate the value of )
Now, by the question
.
and from Step-2, we get
.
Thus,
Therefore, .
Hence option (C) is the correct answer.