Question

in a business meeting each person shakes hands with each other person with the exception of Mr.L. since Mr.L arrives after some people have left, he shakes hands only with those present. if the total number of handshakes is exactly 100, how many people left the meeting before Mr.L arrived?

Answer 1

Dear student
$$\begin{gathered} \mathrm{Let}\mathrm{n}\mathrm{be}\mathrm{the}\mathrm{total}\mathrm{number}\mathrm{of}\mathrm{people}\mathrm{at}\mathrm{the}\mathrm{meeting}\mathrm{before}\mathrm{Mr}.\mathrm{L}\mathrm{arrived}.\mathrm{Then} \\ \mathrm{the}\mathrm{number}\mathrm{of}\mathrm{handshakes}\mathrm{will}\mathrm{be}\frac{\mathrm{n}(\mathrm{n}-1)}{2}.\mathrm{Suppose}\mathrm{m}\mathrm{people}\mathrm{left}\mathrm{before} \\ \mathrm{Mr}.\mathrm{L}\mathrm{arrived}.\mathrm{So}\mathrm{he}\mathrm{shakes}\mathrm{n}-\mathrm{m}\mathrm{hands}.\mathrm{Then} \\ \frac{\mathrm{n}(\mathrm{n}-1)}{2}+\mathrm{n}-\mathrm{m}=100 \\ {\mathrm{n}}^2+\mathrm{n}-2\mathrm{m}=200 \\ \mathrm{m}=\frac{\mathrm{n}(\mathrm{n}+1)}{2}-100. \\ \mathrm{But}0<\mathrm{m}<\mathrm{n}(\mathrm{assuming}\mathrm{that}\mathrm{at}\mathrm{least}\mathrm{one}\mathrm{person}\mathrm{left}).\mathrm{So}\mathrm{we}\mathrm{want} \\ 0<\frac{\mathrm{n}(\mathrm{n}+1)}{2}-100<\mathrm{n} \\ 200<\mathrm{n}(\mathrm{n}+1)<2\mathrm{n}+200. \\ \mathrm{The}\mathrm{first}\mathrm{inequality}\mathrm{suggest}\mathrm{n}\ge 14\mathrm{and}\mathrm{second}\mathrm{inequality}\mathrm{suggest}\mathrm{n}\le 14\Rightarrow \mathrm{n}=14 \\ \mathrm{So},\mathrm{m}=5 \end{gathered}$$
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