Last Monday N female executives (N1) received M male managers (M1)

Question

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

thyagi · Answer

The question is simply asking what are the values of m and n... (Both are integers)..

Statement 1 is clearly not sufficient as m and n can take numerous values...

Statement 2 -
M(m + 2n) = 65
Now factors of 65 include 1,5,13,65...

(1,65) are ruled out since both m and n are greater than 1..

M(m+2n) = 5*13
M = 5 , 5+2n = 13 
N = 4... Bingo... That's all what we need.... (We cannot take m as 13 because taking m as 13 would give n a negative value).

Statement 2 individually is sufficient to answer the question..

Posted from my mobile device

fskilnik · Answer

m,n\,\, \ge \,\,2\,\,\,{m{ints}}\,\,\,\,\left( * ight)

? = C\left( {m + n,2} ight) - C\left( {n,2} ight) = {{\left( {m + n} ight)\left( {m + n - 1} ight)} \over 2} - {{n\left( {n - 1} ight)} \over 2}

? = \frac{{m\left( {m + n - 1} ight) + nm + n\left( {n - 1} ight) - n\left( {n - 1} ight)}}{2} = \,\,\frac{{m\left( {m + 2n - 1} ight)}}{2}\,\,\,\,\, \Leftrightarrow \,\,\,\,\boxed{\,? = \frac{{m\left( {m + 2n - 1} ight)}}{2}\,}

\left( 1 ight)\,\,m < 11\,\,\,\left\{ \matrix{
 \,{m{Take}}\,\,\left( {m,n} ight) = \left( {2,2} ight)\,\,\,\, \Rightarrow \,\,\,{m{?}}\,\,{m{ = }}\,\,{m{5}} \hfill \cr 
 \,{m{Take}}\,\,\left( {m,n} ight) = \left( {2,3} ight)\,\,\,\, \Rightarrow \,\,\,{m{?}}\,\, 
e \,\,{m{5}}\, \hfill \cr} ight.

\left( 2 ight)\,\,m\left( {m + 2n} ight) = 65 = 5 \cdot 13\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * ight)\,\,{m{and}}\,\,\left( {**} ight)} \,\,\,\,\left( {m,m + 2n} ight) = \left( {5,13} ight)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{m{SUFF}}.

\left( {**} ight)\,\,\,m > m + 2n\,\,\,\,\, \Rightarrow \,\,\,n < 0\,\,\,\,\,\,{m{impossible}}

The correct answer is (B).

We follow the notations and rationale taught in the  GMATH  method.

Regards, 
Fabio.

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Last Monday N female executives (N>1) received M male managers (M>1)

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