Question

How many $5$ digit numbers can be formed from digits $0,1,2,3,4,5$?

Answer 1

Step 1: $5$ Digit number without Repetition

If $0$ is in unit place remaining $4$ places can be filled in

$5!$ ways$=5\times 4\times 3\times 2$ ways

$=120$ ways

If $0$ is not in unit's place, the unit's place can be filled with $2$ or $4$ in two ways.

The first place cannot be filled with $0$.

Therefore it can be filled in $4$ ways.

Similarly the remaining places can be filled in $4,3,2$ ways respectively.

Therefore, Number of ways $=4\times 4\times 3\times 2\times 2$

$=192$ways

Therefore, Total number of ways $=120+192$

$=312$ ways

Step 2: $5$ Digit Number with Repetition

If $0$ is filled in units place then

the number of ways $=5\times 6\times 6\times 6\times 1$

$=1,080$ ways.

If $0$ is not filled in units place then

the number of ways$=5\times 6\times 6\times 6\times 2$

$=10\times 216$

$=2,160$ ways

Total number of ways$=1,080+2,160$

$=3,240$ways

Hence, the total count of $5$ digit numbers that can be formed from digits $0,1,2,3,4,5$

(i) without repetition is $312$ ways

(ii) with repetition is $3240$ ways