Question

Ребята, умоляю, помогите мне с этим адским заданием(((
Весь день мучаюсь:((
Заранее огромнейшее спасибо!
(Только без злых шуток(()

Answer 1

$\displaystyle 4).. \frac{5!}{7!}= \frac{5!}{5!*6*7}= \frac{1}{42} \\ \\ \frac{6!}{4!}= \frac{4!*5*6}{4!}=30 \\ \\ \frac{5!}{3!*4!}= \frac{4!*5}{2*3*4!}= \frac{5}{6} \\ \\ \frac{8!}{4!*4!}= \frac{4!*5*6*7*8}{4!*2*3*4}= \frac{1680}{24}=70 \\ \\ \frac{12!}{5!*7!}= \frac{7!*8*9*10*11*12}{7!*2*3*4*5}= \frac{95040}{120}=792 \\ \\ \frac{100!}{98!*2!}= \frac{98!*99*100}{98!*2}=4950$

$\displaystyle 5).. \frac{1}{2!}= \frac{1}{2}= \frac{60}{120};; \frac{1}{5!}= \frac{1}{120} \\ \\ \frac{1}{3!}= \frac{1}{2*3}= \frac{1}{6}= \frac{4}{24} ;; \frac{1}{4!}= \frac{1}{2*3*4}= \frac{1}{24} \\ \\ \frac{1}{7!}= \frac{1}{5!*6*7}= \frac{1}{5!*42};; \frac{1}{5!}= \frac{42}{5!*42} \\ \\ \frac{1}{100!}= \frac{1}{99!*100};; \frac{1}{99!}= \frac{100}{99!*100}$

$\displaystyle 6)..\frac{1}{2!}-\frac{1}{3!}=\frac{1}{2}-\frac{1}{6}=\frac{3}{6}-\frac{1}{6}=\frac{2}{6}=\frac{1}{3};\\\\\frac{1}{3!}-\frac{1}{4!}=\frac{1}{6}-\frac{1}{24}=\frac{3}{24}=\frac{1}{8};\\\\\frac{1}{4!}-\frac{1}{5!}=\frac{5}{4!*5}-\frac{1}{4!*5}=\frac{4}{5!}=\frac{4}{120}=\frac{1}{30}\\\\\frac{1}{5!}-\frac{1}{6!}=\frac{6}{5!*6}-\frac{1}{5!*6}=\frac{5}{6!}=\frac{5}{720}=\frac{1}{144}$

$\displaystyle \frac{1}{n!}-\frac{1}{(n+1)!}=\frac{n+1}{n!(n+1)}-\frac{1}{n!(n+1)}= \frac{n}{n!(n+1)}= \frac{n}{(n+1)!}$