Question

Sa se verifice egalitatea
Combinari de 1000 luate cate 28 = combinari de 999 luate cate 28 + combinari de 999 luate cate 27

Answer 1

Salut,

$C_{1000}^{28}=\dfrac{1000!}{28!\cdot(1000-28)!}=\dfrac{999!}{28!\cdot(1000-28)!}\cdot1000=\\\\=\dfrac{999!}{28!\cdot(1000-28)!}\cdot[(1000-28)+28]=\dfrac{999!\cdot(1000-28)}{28!\cdot(1000-28)!}+\\\\+\dfrac{999!\cdot28}{28!\cdot(1000-28)!}=\dfrac{999!}{28!\cdot(1000-28-1)!}+\dfrac{999!}{27!\cdot(1000-28)!}=\\\\=\dfrac{999!}{28!\cdot(999-28)!}+\dfrac{999!}{27!\cdot(999-27)!}=C_{999}^{28}+C_{999}^{27}.$

Green eyes.