Question

Упростить выражения :

Answer 1

$A^6_{m}+A_{m}^5=m\cdot (m-1)\cdot (m-2)\cdot (m-3)\cdot (m-4)\cdot (m-5)+\\\\+m\cdot (m-1)\cdot (m-2)\cdot (m-3)\cdot (m-4)=\\\\=m\cdot (m-1)\cdot (m-2)\cdot (m-3)\cdot (m-4)\cdot (m-5+1)=\\\\=m\cdot (m-1)\cdot (m-2)\cdot (m-3)\cdot (m-4)^2\\\\\\C_{m}^4\cdot P_4=\frac{m\cdot (m-1)\cdot (m-2)\cdot (m-3)}{m!}\cdot 4!=4!\cdot \frac{m\cdot (m-1)\cdot (m-2)\cdot (m-3)}{1\cdot 2\cdot 3\cdot ...\cdot (m-3)\cdot (m-2)\cdot (m-1)\cdot m}=\\\\=\frac{4!}{1\cdot 2\cdot 3\cdot ...\cdot (m-4)}=\frac{4!}{(m-4)!}$

$\frac{A^6_{m}+A_{m}^5}{C_{m}^{4}\cdot P_4}=\frac{m\cdot (m-1)\cdot (m-2)\cdot (m-3)\cdot (m-4)^2}{\frac{4!}{(m-4)!}}=\\\\=\frac{1}{4!}\cdot m\cdot (m-1)\cdot (m-2)\cdot (m-3)\cdot (m-4)^2\cdot (m-4)!=\\\\=\frac{1}{4!}\cdot \underbrace{m\cdot (m-1)\cdot (m-2)\cdot (m-3)\cdot (m-4)}_{(m-4)!}\cdot (m-4)\cdot (m-4)!=\\\\=\frac{1}{4!}\cdot ((m-4)!)^2\cdot (m-4)$