Answer
Verified
388.5k+ views
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.
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.
Recently Updated Pages
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE
Select the correct plural noun from the given singular class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
The sum of three consecutive multiples of 11 is 363 class 7 maths CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
How many squares are there in a chess board A 1296 class 11 maths CBSE