(X+1)(x+2)(x+3)........(x+9)(x+10)的x^10,x^9,x^8係數為何。 要有算式謝謝(^ з^)?

(1)
x^10 的係數
= 1*1*.....*1
= 1 ..... Ans

(2)
x^9 的係數
= 1 + 2 + ..... + 10
= (1+10)*10/2
= 55 ..... Ans

(3)
(x+1)(x+2)(x+3)........(x+9)(x+10)
= (x+1)(x+10) * (x+2)(x+9) * (x+3)(x+8) * (x+4)(x+7) * (x+5)(x+6)
= ( x^2 + 11x + 10 )( x^2 + 11x + 18 )( x^2 + 11x + 24 )( x^2 + 11x + 28 )( x^2 + 11x + 30 )

x^8 項
= ( 10 + 18 + 24 + 28 + 30 )*(x^2)^4 + C(5,2)*(11x)^2*(x^2)^3
= 110x^8 + 10*121*x^8

x^8 的係數
= 110 + 10*121
= 1320 ...... Ans

Sol
x^10係數
=1
x^9係數
=1+2+3+…+10
=55
x^8係數
=Σ(m=1 to 9)_[Σ(n=m+1 to 10)_mn]
=Σ(m=1 to 9)_｛m[Σ(n=m+1 to 10)_n]｝
=Σ(m=1 to 9)_｛m[Σ(n=1 to 10)_n－Σ(n=1 to m)_n]
=Σ(m=1 to 9)_｛m[10*11/2－m(m+1)/2]｝
=Σ(m=1 to 9)_(55m－m^2/2－m^3/2)
=55Σ(m=1 to 9)_m－(1/2)Σ(m=1 to 9)_m^2－(1/2)m^3
=55*9*10/2－(1/2)*9*10*19/6－(1/2)*81*100/4
=2475－142.5－1012.5
=1320