A student is allowed to select at most n books from a collection of (2n + 1) books. If the total number of ways in which be can select a book is 255, then the value of n equals to

Question

A student is allowed to select at most n books from a collection of (2n + 1) books. If the total number of ways in which be can select a book is 255, then the value of n equals to (A) 6 (B) 5 (C) 4 (D) 3

Tardigrade Admin · Accepted Answer

By the given condition 2n+1C1+ 2n+1C2+ ..........+ 2n+1C4=255 Now 2n+1C1= 2n+1C2n [∵ nCr= nCn-r] 2n+1C2= 2n+1C2n-1 2n+1Cn = 2n+1Cn+1 Adding these, we get 2n+1C0+ 2n+1C1+ 2n+1C2+...........+ 2n+1Cn =2n+1Cn+1+ 2n+1C2n+ 2n+1C2n+1 ⇒ 2[2n+1C0+ 2n+1C1+..........+ 2n+1Cn] =22n+1 ⇒ 2n+1C0+ 2n+1C1+ ..........+ 2n+1Cn=22n ⇒ 2n+1C1+.........+2n+1Cn=22n-1 ⇒ 255=22n-1 ⇒ 256=2n ⇒ 28=22n ⇒ 2n=8 ⇒ n=4. [∵ displaystyle ∑r=0n nCr=2n]

Q. A student is allowed to select at most $n$ books from a collection of $(2 n + 1)$ books. If the total number of ways in which be can select a book is $255$ , then the value of $n$ equals to

$6$

$5$

$4$

$3$

Q. A student is allowed to select at most n books from a collection of (2n+1) books. If the total number of ways in which be can select a book is 255, then the value of n equals to

6

5

4

3

Q. A student is allowed to select at most $n$ books from a collection of $(2 n + 1)$ books. If the total number of ways in which be can select a book is $255$ , then the value of $n$ equals to

$6$

$5$

$4$

$3$