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

Given: Available digits are $1,2,3$ and $4$,

Now, all possible numbers of the 4-digit greater than $4321$ can be formed is as follows:

By using the fundamental principle of multiplication, total 4-digit numbers are

$=4\times 4\times 4\times 4=256$

[Since digits can repeat]

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

By using the fundamental principle of multiplication, number of ways

$=1\times 2\times 4\times 4=32$

But, $4311,4312,4313,4314,4321$ (i.e., $5$ numbers) are less than or equal to $4321$ .

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

$=229$