Question

Вычислите :А2 - С5

8 8
Можно и второе)

Answer 1

$1) \ A_{8} ^{2} -C_{8} ^{5} =\dfrac{8!}{2!} -\dfrac{8!}{5!\cdot(8-5)!} =\dfrac{2!\cdot3\cdot4\cdot5\cdot6\cdot7\cdot8}{2!} -\dfrac{5!\cdot6\cdot7\cdot8}{5!\cdot3!} =\\\\\\=3\cdot4\cdot5\cdot6\cdot7\cdot8-\dfrac{6\cdot7\cdot8}{1\cdot2\cdot3} =3\cdot4\cdot5\cdot6\cdot7\cdot8-7\cdot8=20104$

$2) \ \dfrac{P_{n+2} }{(n+2)\cdot P_{n-2} } =\dfrac{(n+2)!}{(n+2)\cdot(n-2)!} =\dfrac{(n-2)!\cdot(n-1)\cdot n\cdot (n+1)\cdot(n+2)}{(n+2)\cdot(n-2)!} =\\\\\\=(n-1)\cdot n\cdot(n+1)=(n^{2} -1)\cdot n=\boxed{n^{3}-n}$

Answer 2

1.

   $A^2_8-C^5_8=\frac{8!}{(8-2)!}-\frac{8!}{(8-5)!5!}=\frac{8!}{6!}-\frac{8!}{3!5!}=$

$=\frac{1*2*3*4*5*6*7*8}{1*2*3*4*5*6}-\frac{1*2*3*4*5*6*7*8}{(1*2*3)*(1*2*3*4*5)}=\frac{56}{1}-\frac{56}{1}=0$

2.

     $\frac{P_{\pi +2} }{(\pi +2)P_{\pi -2} } =\frac{1*2*3*...*(\pi -2)*(\pi -1)*\pi *(\pi +1)*(\pi +2)}{(\pi +2)*1*2*3*...*(\pi -3)*(\pi -2)}=$

 $=\frac{(\pi -1)*\pi *(\pi +1)}{1}= \pi (\pi ^2-1)=\pi ^3-\pi$