Question

How many four digit numbers that are divisible by $4$ can be formed using the digits $0$ to $7$ if no digit is to occur more than once in any number?

• A. $520$
• B. $370$
• C. $345$
• D. $260$

Answer 1

The correct option is B $370$

We need to find $4$ digit numbers that are divisible by $4$ $\xi$ contain numbers $0$ to $7.$

Since the number is to be divisible by $4$ the last $2$ digits can only be one of the following:

$1).12,16,24,32,36,52,56,64,72,76$

$2).20,40,60,04$

For set $1).$ There are $10$ possible ways for last $2$ digits. For the first digit $0$ cannot be used and also the $2$ digits already used for last $2$ digits. So we have $(8-3)=5$ choices for $1st$ digit and then again $5$ remaining choices for $2nd$ digit.

$\Rightarrow$ Total $=10\times 5\times 5$

$=250.$

For set $2).$ Last $2$ digits can be chosen from $4$ possibilities. First $2$ digits can be chosen from remaining $6$ numbers.

$\Rightarrow$ Total $=4\times {}^6{P}_2$

$=4\times 30$

$=120.$

$\Rightarrow$ Required $=250+120$

$=370.$

Hence, the answer is $370.$