Question

The number of four digit numbers that can be formed from the digits $0,1,2,3,4,5$ with at least one digit repeated is

• A. $420$
• B. $560$
• C. $780$
• D. None of these

Answer 1

The correct option is C $780$

The thousand's place cannot be filled by $0$, so the number of ways to fill the thousand's place is $5$.

The remaining $3$ places can be filled by $6$ numbers in $6^3$ ways (as numbers can be repeated).

Thus the number of $4$ digit numbers $=6^3\times 5=1080$

The number of $4$ digit numbers that do not contain any repeated digits $={{}^{5}p}_1\times {{}^{5}p}_3$ (as thousand's place is to be filled by one of $1,2,3,4,5$ and remaining $3$ places are to be filled by $3$ of the remaining $5$ digits including $0$).

But equation $1$ contains numbers which contain no repeated digits as well as which contain at least one repeated digit.

The number of $4$ digit numbers which contain atleast one repeated digit $=1080-{{}^{5}p}_1\times {{}^{5}p}_3=780$