Question

The number of five digit numbers divisible by $5$ that can be formed using the numbers $0,1,2,3,4,5$ without repetition is

• A. $240$
• B. $216$
• C. $120$
• D. $96$

Answer 1

The correct option is B $216$

The numbers are divisible by $5$ is they end with either $0$ or $5$

Case 1:

Let the number ends with $0$ i.e. unit's digit is $0$

Now there are $5$ choices for ten's place

As the repetition is not allowed, $4$ choices for hundred' place.

$3$ choices for thousand's place and $2$ choices for the first digit.

So, the $5$ digits numbers that can be formed which ends with $0$ is $5\times 4\times 3\times 2=120$

Case 2:

Let the number ends with $5$ i.e. unit's digit is $5$

As the repetition is not allowed, $4$ choices for hundred' place.

$3$ choices for thousand's place and $2$ choices for the first digit.

So, the $5$ digits numbers that can be formed which ends with $5$ is $5\times 4\times 3\times 2=120$

But the first digit can't be $0$. So, we need to subtract that numbers which contains $0$ as first digit.

If $0$ is the first digit, then number of places left to be filled is $4$ and that can be done in $4!$ ways.

So, the $5$ digits numbers that can be formed which ends with $5$ is $120-4!=96$

Thus, the total number of five digit numbers divisible by $5$ is $120+96=216$