Question

Let $k$ be an integer such that the sum of the digits of $k$ is $4$ and ${10}^5<k<{10}^6$. How many values can $k$ have?

• A. $36$
• B. $46$
• C. $45$
• D. $56$

Answer 1

The correct option is D $56$

Given ${10}^5<k<{10}^6$

$\Rightarrow$ $k$ is a $6$ digit number whose digits sum $=4$

Different possibilities for $6$ digits sum $=4:$

$1).(4,0,0,0,0,0)=1$ way

$2).(1,3,0,0,0,0)={}^2{C}_1\times {}^5{C}_1=10$ ways

$3).(2,2,0,0,0,0)={}^3{C}_1\times {}^5{C}_2=3\times 10=30$ ways

$4).(2,1,1,0,0,0)=1\times {}^5{C}_3=10$ ways

$\Rightarrow$ Total $=1+10+5+30+10$

$=56.$

Hence, the answer is $56.$