Question

The sum of all the numbers of four different digits that can be made by using the digits 0,1,2 and 3 is

• A. 26664
• B. 39996
• C. 38664
• D. none of these

Answer 1

The correct option is C 38664

No. of numbers that can be made from $4$ different digits $0,1,2,3=$ $4!-3!=18$
$3!$ is subtracted to remove the numbers in which $0$ comes in the beginning.
no. of times $0$ comes in "last,2nd last and 2nd" position$=3!\times 1=6$
Hence the no. of times rest digits come in "last,2nd last and 2nd" position$=\frac{18-6}{3}=4$
Also the no. of times $0$ comes on the first place$=0$
and no. of times rest digits come on first place $=\frac{18}{3}=6$
Now we will add the digits in each column.
so the "last,2nd last and 2nd" column addition$=0\times 6+4(1+2+3)=24$
and the 1st column addition$=6(1+2+3)=36$
now digit in the last position of sum$=4$ (with carry 2 since the number was 24)
digit in 2nd last position of sum$=6$ (with carry 2 since the number was $24+2=26$)
digit in the 3rd last position of sum$=6$ (with carry 2 since the number was $24+2=26$)
and digit in the 1st place of sum $=36+2=38$
Therefore, the sum $=38664$
Hence, option 'C' is correct.