Question

Выпишите выражения, равные произведениям:
(n-1)*(n-2)*(n-3)*(n-4)*(n-5)
(1-n)*(n-2)*(n-3)*(n-4)*(n-5) (n-1)*(2-n)*(n-3)*(n-4)*(n-5) (1-n)*(2-n)*(3-n)*(4-n)*(5-n)

Answer 1

1) Полагаю, что n ∈ Z

 

$if n > 5, 0! = 1, n! = 1*2*...*n\\ (n-1)*(n-2)*(n-3)*(n-4)*(n-5) =\ (n-1)!/(n-6)!\\ (1-n)*(n-2)*(n-3)*(n-4)*(n-5) =\ -(n-1)*(n-2)*(n-3)*(n-4)*(n-5) =\-(n-1)!/(n-6)!\\ (n-1)*(2-n)*(n-3)*(n-4)*(n-5) =\ -(n-1)*(n-2)*(n-3)*(n-4)*(n-5) =\ -(n-1)!/(n-6)!\\ (1-n)*(2-n)*(3-n)*(4-n)*(5-n) =\ (-1)^5(n-1)*(n-2)*(n-3)*(n-4)*(n-5) =\ -(n-1)!/(n-6)!$

 

 

 

$if n = 1, 2, 3, 4 or 5\\ 0\\ if n < 0, -n = m, m > 0\\ (n-1)*(n-2)*(n-3)*(n-4)*(n-5) =\ (-m-1)*(-m-2)*(-m-3)*(-m-4)*(-m-5) =\ (-1)^5(m+1)*(m+2)*(m+3)*(m+4)*(m+5) =\ -(m+5)!/m!$

 

$(1-n)*(n-2)*(n-3)*(n-4)*(n-5) =\ -(n-1)*(n-2)*(n-3)*(n-4)*(n-5) =\ (-1)^6(m+1)*(m+2)*(m+3)*(m+4)*(m+5) =\ (m+5)!/m!\\ (n-1)*(2-n)*(n-3)*(n-4)*(n-5) =\ (-1)^6(m+1)*(m+2)*(m+3)*(m+4)*(m+5) =\ (m+5)!/m!\\ (1-n)*(2-n)*(3-n)*(4-n)*(5-n) =\ (-1)^5(n-1)*(n-2)*(n-3)*(n-4)*(n-5)=\ (-1)^{10}(m+1)*(m+2)*(m+3)*(m+4)*(m+5) =\ (m+5)!/m!$

 

 

$2) (n-1)*(n-2)*(n-3)*(n-4)*(n-5)=\ (n^2-3n+2)(n^2-7n+12)(n-5) =\ (n^2-3n+2)(n^3-12 n^2+47 n-60) =\ n^5 - 12n^4 +47n^3-60n^2-3n^4+36n^3-141n^2+\180n+2n^3-24n^2+94n-120 =\ n^5 - 15n^4 + 85n^3 - 225n^2 + 274n - 120\\ (1-n)*(n-2)*(n-3)*(n-4)*(n-5) =\ -(n-1)*(n-2)*(n-3)*(n-4)*(n-5) =\-n^5+15n^4 - 85n^3 + 225n^2 - 274n +120$

 

 

 

$(1-n)*(2-n)*(n-3)*(n-4)*(n-5) =\ -(n-1)*(n-2)*(n-3)*(n-4)*(n-5) =\-n^5+15n^4 - 85n^3 + 225n^2 - 274n +120\\ (1-n)*(2-n)*(3-n)*(4-n)*(5-n) =\ (-1)^5(n-1)*(n-2)*(n-3)*(n-4)*(n-5) =\-(n-1)*(n-2)*(n-3)*(n-4)*(n-5) =\-n^5+15n^4 - 85n^3 + 225n^2 - 274n +120$