Question

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

Answer 1

A = {1;2;...;n}
$C_n^2 = 120$
$C_n^2 = \frac{n!}{2!(n-2)!} = \frac{n(n-1)(n-2)!}{2(n-2)!} = \frac{n(n-1)}{2}$
$\frac{n(n-1)}{2} = 120$
$n^2-n-240 = 0$
Δ = 1 + 960 = 961
n₁ = $\frac{1+ 31 }{2}$ = 16 ∈ IN
n₂ = $\frac{1- 31 }{2}$ = -15 ∉ IN
n∈{16}