For a scholarship,at most n candidates out of | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) The number of ways to select at least one candidate is given by $\sum_{k=1}^{n} {^{2n+1}C_k} = 63$. We know that $\sum_{k=0}^{2n+1} {^{2n+1}C_k} =

Vedclass · Accepted Answer

$3$

Vedclass · Answer

(B) The number of ways to select at least one candidate is given by $\sum_{k=1}^{n} {^{2n+1}C_k} = 63$. We know that $\sum_{k=0}^{2n+1} {^{2n+1}C_k} = 2^{2n+1}$. Since ${^{2n+1}C_k} = {^{2n+1}C_{2n+1-k}}$,we have $\sum_{k=0}^{n} {^{2n+1}C_k} = \sum_{k=n+1}^{2n+1} {^{2n+1}C_k} = \frac{2^{2n+1}}{2} = 2^{2n}$. Thus,$\sum_{k=1}^{n} {^{2n+1}C_k} = (\sum_{k=0}^{n} {^{2n+1}C_k}) - {^{2n+1}C_0} = 2^{2n} - 1$. Given $2^{2n} - 1 = 63$,we get $2^{2n} = 64 = 2^6$. Therefore,$2n = 6$,which implies $n = 3$. The maximum number of candidates that can be selected is $n = 3$.

For a scholarship,at most $n$ candidates out of $2n+1$ can be selected. If the number of different ways of selecting at least one candidate for the scholarship is $63$,then the maximum number of candidates that can be selected for the scholarship is -

Similar Questions

In how many ways can $5$ letters be selected from the letters of the word $INDEPENDENT$?

The number of ways to select $n$ objects from $2n$ objects,of which $n$ are identical and the rest are different,is:

If $1 \times 1! + 2 \times 2! + 3 \times 3! + \ldots + n \times n! = 11! - 1$,then the maximum value of ${}^n C_r$ is

In how many ways can a committee of $5$ members be formed from $4$ gentlemen and $6$ ladies such that the number of gentlemen is greater than the number of ladies?

$A$ committee of $12$ members is to be formed from $9$ women and $8$ men. The number of committees in which the women are in majority is

Vedclass Test Series

Exam Paper Generator

Online Exam Module