已知集合M={X/1小于等于X小于等于10,X属于N},对它的非空子集A,将A中每个元素K都乘以（-1）K次方在求和,如：A={1,3,6},可求得和为（-1）X1+(-1)3次方X3+(-1)6次方

题目

已知集合M={X/1小于等于X小于等于10,X属于N},对它的非空子集A,将A中每个元素K都乘以（-1）K次方在求和,如：A={1,3,6},可求得和为（-1）X1+(-1)3次方X3+(-1)6次方X6=2,则对M的所有非空子集,求这些和的总和.

答案

M={x| 1≤x≤10,x∈N }
= { 1,2,3,4,5,6,7,8,9 }
number of non empty subset of M = 2^9 -1
= 511
9C1+9C2+9C3+...+9C9 = 511
number of elements in all those non-empty subset of M
= (1)9C1 + (2)9C2+(3)9C3+...+(9)9C9
= 9 + 2(36) + 3(84)+4(126)+5(126)+6(84)+7(36)+8(9) +9(1)
= 9 +72+ 252+ 504+ 630+ 504+ 252+72+9
= 2304
Number of "1" appears in all those non-empty subet
=Number of "2" appears in all those non-empty subet
=...
=Number of "9" appears in all those non-empty subet
=2304/9 = 256
和的总和
=256( -1+2 - 3 +4 -5+6-7+8-9)
= 256(-5)
= -1280