Question

Sa se determine n∈Ν* pentru care multimea {1,2,...,n} are exact 120 de submultimi cu doua elemente.

Answer 1

$C_n^2=120\\\\ \dfrac{A_n^2}{P_2}=120\\\\ \dfrac{n(n-1)}{1\cdot 2}=120\\\\ n(n-1)=2\cdot 120\\\\ n^2-n=240\\\\ n^2-n-240=0\\\\ n_{12}=\dfrac{1\pm\sqrt{1+4\cdot 240}}{2}=\dfrac{1\pm\sqrt{961}}{2}= \dfrac{1\pm31}{2}\\\\ n_1=\dfrac{1+31}{2}=\dfrac{32}{2}=16\\\\ n_2=\dfrac{1-31}{2}=\dfrac{-30}{2}=-15~\notin N \\\\ \Longrightarrow~~\boxed{\bf n=16}$