Question

Number of natural numbers not exceeding 4321 can be formed with the digits 1,2,3,4 if repetition is allowed

• A. 123
• B. 113
• C. 313
• D. 222

Answer 1

The correct option is C 313

Case $1:$ Four- digit number

Total number of ways in which the $4$ digit number can be formed $=4\times 4\times 4\times 4=256$

Now, the number of ways in which the $4-$ digit numbers greater than $4321$ can be formed is a s follows :

Suppose, the thousand's digit is $4$ and hundred's digit is either $3$ or $4$.

$\therefore$ Number of ways $=2\times 4\times 4=32$

But $4311,4312,43413,4314,4321$ are less than or equal to $4321$

$\therefore$ Remaining number of ways $=256-(32-5)=229$

Case $2:$ Three- digit number

The hundred's digit can be filled in $4$ ways.

Similarly, the ten's digit and the units digit can also be filled in $4$ ways each.

This is because the repetition of digits is allowed.

$\therefore$ Total number of three- digit number $=4\times 4\times 4=64$

Case $3:$ Two- digit number

The ten's digit an the unit's digit can be filled in $4$ ways each. This is because the repetition of digits is allowed.

$\therefore$ Total number of two digits numbers $=4\times 4=16$

Case $4:$ One- digit number

Single digit number can only be $4$

$\therefore$ Required numbers $=229+64+16+4=313$