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CEO  Joined: 15 Aug 2003
Posts: 3223
For a scholarship, at most n candidates out of 2n + 1 can be  [#permalink]

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Question Stats: 100% (03:44) correct 0% (00:00) wrong based on 7 sessions

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For a scholarship, at 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:

1. 3
2. 4
3. 2
4. 5
Intern  Joined: 17 Sep 2003
Posts: 19
Location: GMAT Maze, Chaos.

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Do u know the answer to the problem ?

I m not able to solve it, the qn is kinda confusing.
Intern  Joined: 28 Aug 2003
Posts: 33
Location: USA

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I could do this only by backsolving.

63 = C(2n+1,1) + C(2n+1,2) + .... + C(2n+1,n)

Question stem is asking for n.

A. n = 3 then 2n+1 = 7.

7C1 + 7C2 + 7C3 = 7 + 21 + 35 = 63.

There has to be a better way, I think. Anyone? Vicks? Praet?
CEO  Joined: 15 Aug 2003
Posts: 3223

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edealfan wrote:
I could do this only by backsolving.

63 = C(2n+1,1) + C(2n+1,2) + .... + C(2n+1,n)

Question stem is asking for n.

A. n = 3 then 2n+1 = 7.

7C1 + 7C2 + 7C3 = 7 + 21 + 35 = 63.

There has to be a better way, I think. Anyone? Vicks? Praet?

Honestly, i cant think of a better way.
Vicks , can you help with this.
Manager  Joined: 23 May 2013
Posts: 93
Re: For a scholarship, at most n candidates out of 2n + 1 can be  [#permalink]

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Praetorian wrote:
For a scholarship, at 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. 2
D. 5

No of ways to select at least one candidate out of x candidates is 2^x - 1
2^x - 1 =63
2^x = 64
hence X= 6

But we are told that 2n+1 = x
hence n = 2.5 or >2. Hence we choose A
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Joined: 16 Oct 2010
Posts: 9701
Location: Pune, India
Re: For a scholarship, at most n candidates out of 2n + 1 can be  [#permalink]

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ankur1901 wrote:
Praetorian wrote:
For a scholarship, at 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. 2
D. 5

No of ways to select at least one candidate out of x candidates is 2^x - 1
2^x - 1 =63
2^x = 64
hence X= 6

But we are told that 2n+1 = x
hence n = 2.5 or >2. Hence we choose A

This is not correct. Getting a fractional value for n should be the clue since n = 2.5 could mean n = 2 candidates.

Remember that at most n candidates can be selected. 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.

$$(2n+1)C0 + (2n+1)C1 + (2n+1)C2 + ... + (2n+1)C(n) + (2n+1)C(n+1) + (2n+1)C(n+2) + ... + (2n+1)C(2n+1) = 2^{2n+1}$$

Note that you are given that $$(2n+1)C1 + (2n+1)C2 + ... + (2n+1)C(n) = 63$$. We also know $$(2n+1)C0 = 1.$$
Also recall that $$(2n+1)C0 + (2n+1)C1 + (2n+1)C2 + ... + (2n+1)C(n) = (2n+1)C(n+1) + (2n+1)C(n+2) + ... + (2n+1)C(2n+1)$$
since nCr = nC(n-r)

$$1 + 63 + 63 + 1 = 2^{2n+1}$$
$$n = 3$$

For non-engineering students, this can be solved by backsolving though that's usually time consuming so I doubt GMAT will test such a concept.
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Senior Manager  P
Joined: 10 Apr 2018
Posts: 267
Location: United States (NC)
For a scholarship, at most n candidates out of 2n+1 can be selected  [#permalink]

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For a scholarship, at 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) 5
(D) 6
(E) 8
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