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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

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GMAT Prep

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.

OA D.

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AAPL wrote:
GMAT Prep

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.
Statement 1:
If NONE of the members sells at least 2 tickets, then the 3 members sell a maximum of 1 ticket each, yielding the following total:
1+1+1 = 3.
Too small, since the total number of tickets sold = 6.
Implication:
For a total of 6 tickets to be sold, one of the 3 members must sell at least 2 tickets.
SUFFICIENT.

Statement 2:
LEAST possible case for the 3 members:
0 ticket, 1 ticket, 2 tickets.
The least possible case includes one member selling 2 tickets.
Implication:
For no two members to sell the same number of tickets, one of the 3 members must sell at least 2 tickets.
SUFFICIENT.

The correct answer is D.

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AAPL wrote:
GMAT Prep

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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Portuguese-speakers :: https://www.gmath.com.br

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