GMATH practice exercise (Quant Class 18)
Last Monday N female executives (N>1) received M male managers (M>1) for a business meeting. If every person shook hands exactly once with every other person in the meeting, what is the difference between the total number of shaking hands and the number of shaking hands among the female executives only?
(1) M < 11
(2) M(M+2N) = 65
Answer: [spoiler]_____(B)__[/spoiler]
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Last Monday N female executives (N>1) received M male man
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$$m,n\,\, \ge \,\,2\,\,\,{\rm{ints}}\,\,\,\,\left( * \right)$$fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 18)
Last Monday N female executives (N>1) received M male managers (M>1) for a business meeting. If every person shook hands exactly once with every other person in the meeting, what is the difference between the total number of shaking hands and the number of shaking hands among the female executives only?
(1) M < 11
(2) M(M+2N) = 65
$$? = C\left( {m + n,2} \right) - C\left( {n,2} \right) = {{\left( {m + n} \right)\left( {m + n - 1} \right)} \over 2} - {{n\left( {n - 1} \right)} \over 2}$$
$$? = \frac{{m\left( {m + n - 1} \right) + nm + n\left( {n - 1} \right) - n\left( {n - 1} \right)}}{2} = \,\,\frac{{m\left( {m + 2n - 1} \right)}}{2}\,\,\,\,\, \Leftrightarrow \,\,\,\,\boxed{\,? = \frac{{m\left( {m + 2n - 1} \right)}}{2}\,}$$
$$\left( 1 \right)\,\,m < 11\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {2,2} \right)\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\,{\rm{ = }}\,\,{\rm{5}} \hfill \cr
\,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {2,3} \right)\,\,\,\, \Rightarrow \,\,\,{\rm{?}}\,\, \ne \,\,{\rm{5}}\, \hfill \cr} \right.$$
$$\left( 2 \right)\,\,m\left( {m + 2n} \right) = 65 = 5 \cdot 13\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)\,\,{\rm{and}}\,\,\left( {**} \right)} \,\,\,\,\left( {m,m + 2n} \right) = \left( {5,13} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.$$
$$\left( {**} \right)\,\,\,m > m + 2n\,\,\,\,\, \Rightarrow \,\,\,n < 0\,\,\,\,\,\,{\rm{impossible}}$$
The correct answer is (B).
We follow the notations and rationale taught in the GMATH method.
Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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