求(1-x)+(1-x)^2+(1-x)^3+…+(1-x)^50展开式中x^4的系数

来源：学生作业学帮网编辑：学帮网时间：2024/05/13 13:59:21

x[(1-x)+(1-x)^2+(1-x)^3+…+(1-x)^50]
=[1-(1-x)]*[1+(1-x)+(1-x)^2+(1-x)^3+…+(1-x)^50-1]
=1-(1-x)^51-x
=1-x+(x-1)^51
所以要算原式中x^4的系数,只要算上式中x^5的系数：
C(51,46)*(-1)^46=C(51,5)=2349060

50
-1275 x
+20825 x^2
-249900 x^3
+2349060 x^4
-18009460 x^5
+115775100 x^6
-636763050 x^7
+3042312350 x^8
-12777711870 x^9
+47626016970 x^10
-158753389900 ...

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50
-1275 x
+20825 x^2
-249900 x^3
+2349060 x^4
-18009460 x^5
+115775100 x^6
-636763050 x^7
+3042312350 x^8
-12777711870 x^9
+47626016970 x^10
-158753389900 x^11
+476260169700 x^12
-1292706174900 x^13
+3188675231420 x^14
-7174519270695 x^15
+14771069086725 x^16
-27900908274925 x^17
+48459472266975 x^18
-77535155627160 x^19
+114456658306760 x^20
-156077261327400 x^21
+196793068630200 x^22
-229591913401900 x^23
+247959266474052 x^24
-247959266474052 x^25
+229591913401900 x^26
-196793068630200 x^27
+156077261327400 x^28
-114456658306760 x^29
+77535155627160 x^30
-48459472266975 x^31
+27900908274925 x^32
-14771069086725 x^33
+7174519270695 x^34
-3188675231420 x^35
+1292706174900 x^36
-476260169700 x^37
+158753389900 x^38
-47626016970 x^39
+12777711870 x^40
-3042312350 x^41
+636763050 x^42
-115775100 x^43
+18009460 x^44
-2349060 x^45
+249900 x^46
-20825 x^47
+1275 x^48
-51 x^49
+x^50
呵呵。。。

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