Question

The number of four digit numbers that can be formed 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

Hint: As we see it is a basic question of permutation and combination and we have to make a four digit number by using digits 0,1,2,3,4,5. And we see digits contain 0 so we can not fill 0 at a thousand places and rest we can arrange in such a manner so that number gets repeated.

Complete step-by-step answer:
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 $6 \times 6 \times 6$ ways (as numbers can be repeated).
Thus the number of 4 digit number = $5 \times 6 \times 6 \times 6$
Number of 4 digit number is $1080$
The number of 4 digit numbers that do not contain any repeated digits = ${}^5{P_1} \times {}^5{P_3}$ --- (1)
(a thousand's place is to be filled by one of 1,2,3,4,5 and the 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 at least one repeated digit =$1080 - {}^5{P_1} \times {}^5{P_3}$
Now how to solve permutation
${}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$
${}^5{P_1} = \dfrac{{5!}}{{\left( {5 - 1} \right)!}}$ = $\dfrac{{5 \times 4 \times 3 \times 2 \times 1}}{{4 \times 3 \times 2 \times 1}}$ = $5$
Similarly
${}^5{P_3} = \dfrac{{5!}}{{\left( {5 - 3} \right)!}}$ = $\dfrac{{5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1}}$ = $5 \times 4 \times 3 = 60$
So number of 4 digit numbers which contain at least one repeated digit = $1080 - \left( {5 \times 60} \right)$
$ \Rightarrow 1080 - 300$
$ \Rightarrow 780$
So number of 4 digit numbers which contain at least one repeated digit is $780$
So option C is the correct answer.

Note: As I say in hint we can note take 0 at thousand place because when we take 0 at thousand place it is a three digit number example $0123$ is a three digit number. We can also see that the question talks about at least one digit repeat not all digits repeat these both terms are different.