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
Answers & Comments
chAnjani 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
Answers & Comments
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