Question

How many four digit natural numbers not exceeding $4321$ can be formed with the digits $1,2,3$ and $4$, if the digits can repeat ?

Answer 1

The given digits are $1,2,3$ and $4$.
These digits can be repeated while forming the numbers.
The total number of ways in which the $4$ digit numbers can be formed $=4\times 4\times 4\times 4=256$
Now, first we will find the number of ways in which $4$ digit numbers greater than $4321$ as follow:
Suppose the thousand digit is $4$ and hundred digit is either $3$ or $4$.
Therefore, the number of ways$=2\times 4\times 4=32$
But, the numbers $4311,4312,4313,4314,4321$ are less than or equal to $4321$.
Thus, the out of $32$ numbers, the five number are less than or equal to $4321$.
So, the number of four digit numbers greater than $4321$ are $32-5=27$.
Hence, the number of ways the four digit numbers formed that are not exceeding $4321=$ total number of $4$ digit numbers formed $-$ the number of four digit numbers greater than $4321$.
$=256-27$
$=229$.