Question

The sum of all the $4-$digit distinct numbers that can be formed with the digits $1,2,2$ and $3$ is:

• A. $26664$
• B. $122664$
• C. $122234$
• D. $22264$

Answer 1

The correct option is A $26664$

Digits are $1,2,2$ and $3.$
Total distinct numbers $=\frac{4!}{2!}=12$
There are three distinct numbers when $1$ is at unit place.
There are three distinct numbers when $3$ is at unit place.
There are six distinct numbers when $2$ is at unit place.

So, the sum of all the $4-$digit distinct numbers is,
$=(3+9+12)({10}^3+{10}^2+10+1)$
$=26664$

Alternate Solution:
$1223$
$1232$
$1322$
$3122$
$3212$
$3221$
​​​​​​​​​​​​​​​​​​​​​$2132$
​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​$2312$
​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​$2213$
​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​$2231$
​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​$2321$
​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​$\underset{–}{+2123}$
$26664$