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# Did one of the 3 members of a certain team sell at least 2

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Did one of the 3 members of a certain team sell at least 2  [#permalink]

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Updated on: 25 Jun 2017, 04:59
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Difficulty:

55% (hard)

Question Stats:

57% (00:47) correct 43% (00:51) wrong based on 439 sessions

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Did one of the 3 members of a certain team sell at least 2 raffle tickets yesterday?

(1) The 3 members sold a total of 6 raffle tickets yesterday.
(2) No 2 of the members sold the same number of raffle tickets yesterday.

Acc to (1) one should mean only 1 member sold atleast 2 raffle, NOT atleast one member sold atleast 2 raffle.
Hence, the two contradictory options should be :
0,0,6- one member sold atleast 2 raffle
2,2,2- more than one member sold atleast 2 raffle.

Originally posted by saurabhkhatrinitk on 04 Aug 2012, 09:18.
Last edited by hazelnut on 25 Jun 2017, 04:59, edited 3 times in total.
EDITED THE QUESTION.
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Re: Did one of the 3 members of a certain team sell at least 2  [#permalink]

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04 Aug 2012, 09:40
10
7
Did one of the 3 members of a certain team sell at least 2 raffle tickets yesterday?

The question basically asks whether there is a member who sold at least 2 tickets (so 2 or more).

(1) The 3 members sold a total of 6 raffle tickets yesterday. If each of the 3 members sold less than 2 tickets, then the total # of tickets sold cannot be 6, hence at least one member sold at least 2 tickets. Sufficient.

Or: can we split 6 tickets so that ALL 3 members to have sold less than 2 tickets? No: (6,0,0); (5,1,0), (4,1,1); (4,2,0); (3,3,0),(3,2,1), (2,2,2). Sufficient.

(2) No 2 of the members sold the same number of raffle tickets yesterday. If one member sold 0 tickets and another sold 1 ticket (the least possible numbers), then the third one must have sold more than 1, so 2 or more. Sufficient.

Hope it's clear.
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Re: Did one of the 3 members of a certain team sell at least 2  [#permalink]

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04 Aug 2012, 09:50
Bunuel wrote:
Did one of the 3 members of a certain team sell at least 2 raffle tickets yesterday?

The question basically asks whether there is a member who sold at least 2 tickets (so 2 or more).

(1) The 3 members sold a total of 6 raffle tickets yesterday. If each of the 3 members sold less than 2 tickets, then the total # of tickets sold cannot be 6, hence at least one member sold at least 2 tickets. Sufficient.

Or: can we split 6 tickets so that ALL 3 members to have sold less than 2 tickets? No: (6,0,0); (5,1,0), (4,1,1); (4,2,0); (3,3,0),(3,2,1), (2,2,2). Sufficient.

(2) No 2 of the members sold the same number of raffle tickets yesterday. If one member sold 0 tickets and another sold 1 ticket (the least possible numbers), then the third one must have sold more than 1, so 2 or more. Sufficient.

Hope it's clear.

If the Q asks "whether there is a member who sold at least 2 tickets (so 2 or more)" then it should have been framed as atleast one NOT one which indicates Exactly one. Hence the doubt. Infact, i would say the Q should use the word Atleast or Exactly to make it clear.
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Re: Did one of the 3 members of a certain team sell at least 2  [#permalink]

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04 Aug 2012, 11:44
saurabhkhatrinitk wrote:
Bunuel wrote:
Did one of the 3 members of a certain team sell at least 2 raffle tickets yesterday?

The question basically asks whether there is a member who sold at least 2 tickets (so 2 or more).

(1) The 3 members sold a total of 6 raffle tickets yesterday. If each of the 3 members sold less than 2 tickets, then the total # of tickets sold cannot be 6, hence at least one member sold at least 2 tickets. Sufficient.

Or: can we split 6 tickets so that ALL 3 members to have sold less than 2 tickets? No: (6,0,0); (5,1,0), (4,1,1); (4,2,0); (3,3,0),(3,2,1), (2,2,2). Sufficient.

(2) No 2 of the members sold the same number of raffle tickets yesterday. If one member sold 0 tickets and another sold 1 ticket (the least possible numbers), then the third one must have sold more than 1, so 2 or more. Sufficient.

Hope it's clear.

If the Q asks "whether there is a member who sold at least 2 tickets (so 2 or more)" then it should have been framed as atleast one NOT one which indicates Exactly one. Hence the doubt. Infact, i would say the Q should use the word Atleast or Exactly to make it clear.

Not so (even though I do see why you are confused).

Actually it's opposite, if the question meant exactly (only) one, then it would say so.
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Re: Did one of the 3 members of a certain team sell at least 2  [#permalink]

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29 Nov 2016, 19:45
The question looks kind of tricky to me,
However deep diving I think- the bottomline of the question is did any of the 3 members sold more than 1 ticket.

Now total no of tickets sold in 6 and tickets cannot be negative or fractions hence yes statement 1 is true.

Similarly statement 2 is also true if 3 of them sold different number of tickets then they can sell (0,1, 2) tickets .

Thus both the statements are sufficient to prove it correct.

Hope this helps.
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Re: Did one of the 3 members of a certain team sell at least 2  [#permalink]

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02 Feb 2017, 17:35
statement (1):
there's a statement called the pigeonhole principle, which basically says the following two things:
* if the AVERAGE of a set of integers is an INTEGER n, then at least one element of the set is > n.
* if the AVERAGE of a set of integers is a NON-INTEGER n, then at least one element of the set is > the next integer above n.
this principle is easy to prove: if you assume the contrary, then you get the absurd situation in which every element of a set is below the average of the set. that is of course impossible.

specifically, statement (1) is a case of the first part of the principle: the average of the set is 6/3 = 2, so at least one element of the set must be 2 or more.

statement (2):
there are only two ways not to sell at least 2 tickets: sell 0 tickets, and sell 1 ticket.
if everyone sells a different # of tickets, then you can't fit three people into these two categories.
therefore, someone must have sold at least 2 tickets.

Hence D.
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Re: Did one of the 3 members of a certain team sell at least 2  [#permalink]

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10 Feb 2017, 22:33
ram186 wrote:
Did one of the 3 members of a certain team sell at least 2 raffle tickets yesterday.

(1) The 3 members sold a total of 6 raffle tickets yesterday

(2) No 2 of the members sold the same number of raffle tickets yesterday

(1) 3 members sold a total of 6 raffle tickets - Combinations are 2,2,2, 4,1,1, 5,0,1, and so on. So in each case at least one would've sold more than 2. - Sufficient

(2) No 2 of the members sold the same number of raffle tickets yesterday - We don't know how many tickets were sold yesterday but we know no 2 person sold the same no. of tickets. Since they are tickets they must be 3 different non negative integers. The 2 different smallest non negative integers are 0, and 1. Hence the other one has to be either 2 or greater than 2. So either way we can tell one person has sold at least 2 raffle tickets. - Sufficient

Hope this explains.

Cheers!
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Re: Did one of the 3 members of a certain team sell at least 2  [#permalink]

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11 Feb 2017, 10:54
Sure. This is a good alternative solution.

But will you really remember this principle on the test day ?

In my opinion, it's better to go intuitive about questions. But that's what I think. Everyone's got his/her way of doing things !

anairamitch1804 wrote:
statement (1):
there's a statement called the pigeonhole principle, which basically says the following two things:
* if the AVERAGE of a set of integers is an INTEGER n, then at least one element of the set is > n.
* if the AVERAGE of a set of integers is a NON-INTEGER n, then at least one element of the set is > the next integer above n.
this principle is easy to prove: if you assume the contrary, then you get the absurd situation in which every element of a set is below the average of the set. that is of course impossible.

specifically, statement (1) is a case of the first part of the principle: the average of the set is 6/3 = 2, so at least one element of the set must be 2 or more.

statement (2):
there are only two ways not to sell at least 2 tickets: sell 0 tickets, and sell 1 ticket.
if everyone sells a different # of tickets, then you can't fit three people into these two categories.
therefore, someone must have sold at least 2 tickets.

Hence D.
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Did one of the 3 members of a certain team sell at least 2  [#permalink]

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16 Aug 2017, 02:02
I agree with some people who posted their doubts about the questions earlier. I chose option E.

If the question had clearly mentioned "Did atleast one of the 3 members of a certain team sell at least 2 raffle tickets yesterday." then it would've caused much less confusion.

But at the end of the day it's an official question, so we can't argue much. :-/
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Re: Did one of the 3 members of a certain team sell at least 2  [#permalink]

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31 Oct 2018, 12:35
saurabhkhatrinitk wrote:
Did one of the 3 members of a certain team sell at least 2 raffle tickets yesterday?

(1) The 3 members sold a total of 6 raffle tickets yesterday.
(2) No 2 of the members sold the same number of raffle tickets yesterday.

$$A,B,C\, \ge 0\,\,{\rm{ints}}\,\,\,\left( * \right)$$

$$?\,\,\,:\,\,\,\,A \ge 2\,\,\,{\rm{or}}\,\,\,B \ge 2\,\,\,{\rm{or}}\,\,\,C \ge 2$$

$$\left( 1 \right)\,\,A + B + C = 6\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle$$

$$\left( {**} \right)\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\left\{ \matrix{ \,A \le \,\,1 \hfill \cr \,B \le \,\,1 \hfill \cr \,C \le \,\,1 \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,A + B + C\,\, \le \,\,3\,\,\,\,\, \Rightarrow \,\,\,\,\left( 1 \right)\,\,\,{\rm{contradicted}}$$

$$\left( 2 \right)\,\,A,B,C\,\,{\rm{different}}\,\,\,\,\mathop \Rightarrow \limits^{\left( {***} \right)} \,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle$$

$$\left( {***} \right)\,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\, \Rightarrow \,\,\,\,\left\{ \matrix{ \,A \le \,\,1\,\,\,\, \Rightarrow \,\,\,A \in \left\{ {0,1} \right\} \hfill \cr \,B \le \,\,1\,\,\,\, \Rightarrow \,\,\,B \in \left\{ {0,1} \right\} \hfill \cr \,C \le \,\,1\,\,\,\, \Rightarrow \,\,\,C \in \left\{ {0,1} \right\} \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( 2 \right)\,\,\,{\rm{contradicted}}\,$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: Did one of the 3 members of a certain team sell at least 2  [#permalink]

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16 Nov 2018, 08:29
(1) The 3 members sold a total of 6 raffle tickets yesterday.
Minimum case = 2 each by 3 members
other cases -0,0,6 or 1,0,5 and many more in all at least one has sold 2 tickets

(2) No 2 of the members sold the same number of raffle tickets yesterday.
Minimum case - 0,1,5 or 1,2,3 or 0,2,4 0r any other Tickets will always be at least 2 for one of member as case
2,2,2 is not valid here

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Re: Did one of the 3 members of a certain team sell at least 2 &nbs [#permalink] 16 Nov 2018, 08:29
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