Class 11

Math

Algebra

Permutations and Combinations

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?

- $520$
- $370$
- $345$
- $260$

We need to find $4$ digit numbers that are divisible by $4$ $ξ$ 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.

$⇒$ Total $=10×5×5$

$=250.$

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

$⇒$ Total $=4×_{6}P_{2}$

$=4×30$

$=120.$

$⇒$ Required $=250+120$

$=370.$

Hence, the answer is $370.$