Question

решите примеры
тема факториал

Answer 1

 

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

 

 

 

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

 

 

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

 

 

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