Question

The number of integers between $1$ and $1000000$ that have the sum of the digits $18$ is

• A. $25927$
• B. $25827$
• C. $24927$
• D. None of thes

Answer 1

The correct option is A $25927$

Any number between $1$ and $1000000$ must be of less than $7$ digits. Thus it must be of the form $a_1{a}_2{a}_3{a}_4{a}_5{a}_6$ where $a_1{,a}_2,a_3,a_4,a_5,a_6\in \{0,1,2,\mathrm{3...9}\}$

Given, sum of digits$=18$

so, $a_1{+a}_2,{+a}_3+a_4+a_5+a_6=18$

Required number

= coefficient of $x^{18}$ in ${(1+x+x^2+...x^9)}^6$

= coefficient of $x^{18}$ in ${\left(\frac{1-x^{10}}{1-x}\right)}^6$

= coefficient of $x^{18}$ in ${(1-x^{10})}^6{(1-x)}^{-6}$

= coefficient of $x^{18}$ in $(1-{{}^{6}C}_1{x}^{10}){(1-x)}^{-6}$ (leaving terms having powers of x greater than 18)

= coefficient of $x^{18}$ in ${(1-x)}^{-6}$ - ${{}^{6}C}_1$ .coefficient of $x^8$ in ${(1-x)}^{-6}$

$={{}^{6+18-1}C}_{18}-6.{{}^{6+8-1}C}_{18}={{}^{23}C}_5-6.{{}^{13}C}_8=25927$