Question

5* C n ^ 3 =C n+2 ^ 4


Срочна​

Answer 1

Ответ:

$\boxed{\ C_{n}^{k}=\dfrac{n!}{k!\, (n-k)!}\ }\\\\\\5\cdot C_{n}^3=C_{n+2}^4\\\\\\5\cdot \dfrac{n!}{3!\cdot (n-3)!}=\dfrac{(n+2)!}{4!\cdot (n-2)!}\\\\\\5\cdot \dfrac{(n-3)!(n-2)(n-1)\, n}{3!\cdot (n-3)!}=\dfrac{(n-2)!(n-1)\, n\, (n+1)(n+2)}{4!\cdot (n-2)!}\\\\\\5\cdot \dfrac{(n-2)(n-1)\, n}{3!}=\dfrac{(n-1)\, n\, (n+1)(n+2)}{3!\, 4}\\\\\\5\cdot (n-2)=\dfrac{(n+1)(n+2)}{4}\\\\\\20n-40=n^2+3n+2\ \ ,\ \ \ n^2-17n+42=0\ \ ,\ \ n_1=3\ ,\ n_2=14\ \ (Viet)\\\\\\Otvet:\ \ n_1=3\ ,\ n_2=14\ .$