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.
_________________
Fabio Skilnik ::
GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here:
https://gmath.net