Question

Să se rezolve inecuația $C\binom{2}{n}\ \textless \ 10$,$n \geq 2, n$ natural.

Answer 1

[tex]C(^2_n)=\frac{n!}{2!(n-2)!}=\frac{(n-2)!\cdot(n-1)\cdot n}{2\cdot(n-2)!}=\frac{(n-1)n}{2}\\ Avem\ ca:\\ \frac{(n-1)n}{2}\ \textless \ 10\\ n^2-n-20\ \textless \ 0\\ n^2-5n+4n-20\ \textless \ 0\\ n(n-5)+4(n-5)\ \textless \ 0\\ (n-5)(n+4)\ \textless \ 0\\ Deci\ n\in (-4,5),dar\ cum\ n\in \mathbb{N} si\ n\geq 2 \Rightarrow n\in \{2,3,4\}\\ S=\{2,3,4\}[/tex]