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InterviewSolution
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How many integers between 1 and 1000000 have the sum of the digit equal to 18? |
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Answer» Integers betweenn 1 and 1000000 will be 1,2,3,4,5 or 6 digits and given sum of digit=18 Thus, we need to obtain the number of solutions of the equation `x_(1)+x_(2)+x_(3)+x_(4)+x_(5)+x_(6)=18` . . . (i) where, `0 lt x_(i) le 9, i=1,2,3,4,5,6` Therefore, the number of solutions of Eq. (i) will be =Coefficient of `x^(18)` in `(x^(0)+x^(1)+x^(2)+x^(3)+ . . .+x^(9))^(6)` =Coefficient of `x^(18)` in `((1-x^(10))/(1-x))^(6)` =Coefficient of `x^(18)` in `(1-x^(10))^(6)(1-x)^(-6)` =Coefficient of `x^(18)` in `(1-6x^(10))(1+.^(6)C_(1)x+.^(7)C_(2)x^(2)+ . . .+.^(13)C_(8)x^(8)+ . . .+.^(23)C_(18)x^(18)+ . . .)` `=.^(23)C_(18)-6*.^(13)C_(8)=.^(23)C_(5)-6*.^(13)C_(5)` `=(23*22*21*20*19)/(1*2*3*4*5)-6*(13*12*11*10*9)/(1*2*3*4*5)` `=33649-7722=25927` |
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