Question

a student is allowed to select at most n books from a collection of 2n+1 .if the total no. of ways in which he can select atleast one book is 63.find the value of n

Answer 1

The no. of ways in which one can select at least one book

and at the most n books from 2n+1 books is

2n+1C1+2n+1C2+2n+1C3+...........2n+1Cn

we know

(1+x)2n+1=2n+1C0+2n+1C1.x+........2n+1C2n+1.xn

Put x=1 and also we know that nCr=nCn−r

2n+1C0+2n+1C1+....2n+1Cn+2n+1Cn+1....2n+1C2n+1

=22n+1

i.e.,2[2n+1C0+2n+1C1+....2n+1Cn]=22n+1

⇒2n+1C0+2n+1C1+....2n+1Cn=22n

⇒2n+1C1+....2n+1Cn=22n−1 (Since2n+1C0=1)

Given that2n+1C1+2n+1C2+2n+1C3+...........2n+1Cn=63

⇒22n−1=63

⇒22n=64

⇒n=3