Решите примеры тема факториал

$1) \ a) \frac{n!}{(n-1)!} = \frac{1*2*...*(n-1)*n}{1*2*...*(n-1)} = n\\\\ b) \ \frac{n!}{(n-2)!} = \frac{1*2*...*(n-3)*(n-2)*(n-1)*n}{1*2*...*(n-3)*(n-2)} = (n-1)*n\\\\ c) \ \frac{(n-3)!}{n!} = \frac{1*2*...*(n-3)}{1*2*...*(n-3)*(n-2)*(n-1)*n} = \frac{1}{(n-2)*(n-1)*n}\\\\$

$d) \ \frac{A^3_{n+1}}{A^2_{n}} = \frac{\frac{(n+1)!}{(n+1-3)!}}{\frac{n!}{(n-2)!}} = \frac{(n+1)!(n-2)!}{(n-2)!n!} = \frac{(n+1)!}{n!} = n+1\\\\ e) \ \frac{A^3_{n+1}}{A^2_{n+1}} = \frac{\frac{(n+1)!}{(n+1-3)!}}{\frac{n!}{(n+1-2)!}} = \frac{(n+1)!(n-2)!}{(n-1)!n!} =\\ \frac{(n+1)!}{n!(n-1)} = (n+1)/(n-1) = 1 + \frac{2}{n-1}\\\\$

$2) \ a) \frac{1}{n!} + \frac{1}{(n+1)!} = \frac{1}{1*2*...*n} + \frac{1}{1*2*...*n*n+1} =\\ \frac{n+1}{(n+1)!} + \frac{1}{(n+1)!} = \frac{n+2}{(n+1)!} = \frac{(n+2)!}{(n+1)!(n+1)!}\\\\ b) \ \frac{1}{(n+1)!} - \frac{1}{n!} = \frac{1}{1*2*...*n*n+1} - \frac{1}{1*2*...*n} =\\ \frac{1}{(n+1)!} - \frac{n+1}{(n+1)!} = -\frac{n!}{(n+1)!(n-1)!}$

$3) \ a) \ \frac{10! - 8!}{89} = \frac{8!(9*10 -1)}{89} = 8! = 40320\\\\ b) \ \frac{5! + 6!}{4!} = \frac{4!(5 +5*6)}{4!} = 35\\\\ c) \ A^{m-5}_{m-1} = \frac{(m-1)!}{(m-1-m+5)!} = \frac{(m-1)!}{4!} = \frac{(m-1)!}{24} \\\\$