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How many integers with a total of digits of 18 exist between 1 and 1000000?

Option: 1

25677


Option: 2

33649


Option: 3

7722


Option: 4

25927


Answers (1)

best_answer

 Let the numerals be \mathrm{a_{1}, a_{2}, a_{3}, a_{4}, a_{5}} and \mathrm{a_{6}}.

Any number between 1 and 1000000 will only have seven digits.
Then it must have the following format \mathrm{a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}}.

Where \mathrm{a_{1}, a_{2}, a_{3}, a_{4}, a_{5}} and \mathrm{a_{6}\: \varepsilon\: (0,1,2,3,4,5,6,7,8,9)}
\mathrm{a_{1}+a_{2}+a_{3}+a_{4}+a_{5}=18}, where \mathrm{0 \leq a i \leq 9, i=(0,1,2,3,4,5,6,7,8,9)}

Then the number of integers will be equal to the coefficient of \mathrm{x^{18}} in \mathrm{\left(1+x+x^{2}+\ldots x^{9}\right)^{6}}

Coefficient of \mathrm{x^{18}} in \mathrm{\left(\frac{1-x^{10}}{1-x}\right)^{6}}

Coefficient of \mathrm{x^{18}} in \mathrm{\left(\frac{1-x^{10}}{1-x}\right)^{6}} \mathrm{= }  Coefficient of \mathrm{x^{18} } in  \mathrm{\mathrm{\left[\left(1-{ }^{6} C_{1} x^{10} \ldots \ldots\right)(1-x)^{-6}\right]}} 

Coefficient of  \mathrm{x^{18}} in \mathrm{(1-x)^{-6}-{ }^{6} C_{1} x^{10} \times Coefficient\, of \,x^{8}\: in\; (1-x)^{-6} }

\mathrm{={ }^{6+18-1} C_{5}-6{ }^{6+8-1} C_{5}={ }^{23} C_{5}-6{ }^{13} C_{5}}
\mathrm{33649-7722=25927}

Option (d) is correct.

Posted by

Pankaj

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