CAT 1999 | Question: 60

Question

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

• A. 3
• B. 4
• C. 6
• D. 5

Answer 1

The number of ways you can select at least 1 candidate up to n candidates out of the total 2n+1 is given as 63.  

$^{2n+1}C_1+^{2n+1}C_2+...+^{2n+1}C_n$=63

and 

$^{2n+1}C_0$+$^{2n+1}C_1$+$^{2n+1}C_2$+...+$^{2n+1}C_n$+$^{2n+1}C_n+1$+$^{2n+1}C_n+2$+...+$^{2n+1}C_2n+1$=$2^{2n+1}$  ..............(1)

We know $^{2n+1}C_0=1$     and   $^n{C}_r = ^{n}C_n-r$

So

$^{2n+1}C_0+^{2n+1}C_1+^{2n+1}C_2+...+^{2n+1}C_n=^{2n+1}C_n+1+^{2n+1}Cn+2+...+^{2n+1}C_2n+1$

...........(2)

From(1) and (2)

1+63+63+1=$2^{2n+1}$

$2^{7}$ =$2^{2n+1}$

n=3

Hence,Option(A)3.

Answer 2

Answer D) 5

63 different ways we can select at least 1 candidate

2n+1 =63

n=31

Max no of candidate can select log31 =5

Comments

You did wrong because 2n+1 is total number of student  and  63 is number of different ways of selection for  scholarship.So they can't be equal.

and one more thing how log31 =5 ?