Question

*Determinati n ∈ N* pentru care multimea A={1,2,...,n} are exact 10 submultimi cu doua elemente.

Answer 1

[tex]\text{Numarul de submultimi cu doua elemente este:} \boxed{\binom{2}{n}}\\ \text{Prin urmare avem ca:}\\ \binom{2}{n}=10\\ \dfrac{n!}{2!\cdot (n-2)!}=10\\ \dfrac{(n-2)!\cdot(n-1)\cdot n}{2\codt (n-2)!}=10\\ \dfrac{(n-1)\cdot n}{2}=10\\ n^2-n=20\\ n^2-n-20=0\\ (n-5)(n+4)=0\Rightarrow n_1=5\\ ~~~~~~~~~~~~~~~~~~~~~~~~~\Rightarrow n_2=-4(\text{nu convine})\\ S:n=5 [/tex]