Question

Determinați numărul natural nenul n stiind că mulțimea A={1,2,3...n} are exact 10 submulțimi cu doua elemente .​

Answer 1

Vom folosi combinarile:

$C_n^k=\frac{n!}{k!(n-k)!} \\\\n!=1\cdot2\cdot 3\cdot...\cdot n\\\\k!=1\cdot2\cdot 3\cdot...\cdot k$

$C_n^2=10$

$\frac{n!}{2!\cdot (n-2)!} =10$

Stim ca:

$n!=(n-2)!(n-1)n$

$\frac{(n-2)!(n-1)n}{2!\cdot (n-2)!}=10\\\\ \frac{n(n-1)}{2}=10\\\\ n(n-1)=20\\\\n^2-n-20=0\\\\\Delta=1+80=81\\\\n_1=\frac{1+9}{2} =5\\\\n_2=\frac{1-9}{2} =-4 < 0\ NU$

Raspuns: n=5

Un alt exercitiu asemanator gasesti aici: https://brainly.ro/tema/5012184

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