Question

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

• A. $1480$
• B. $480$
• C. $600$
• D. $300$

Answer 1

The correct option is A $1480$

For numbers to be divisible by $4$, the last two digits should be divisible by $4$.

The numbers formed from digits $0-7$ should have $04,12,16,20,24,32,36,40,52,56,60,64,72,76$ as last two digits to be divisible by $4$.

The total possible cases are $=14$

1) When the last two digits don't have a zero that is $10$ cases

Then the left most digit will have $5$ options since $0$ cannot be at the leftmost digit(otherwise it will become a $4$ digit number)

The next digit will also have $5$ options(including $0$)

The next place will have only $4$ options since digits cannot be repeated.

Hence the number of possible ways are $=10\times 5\times 5\times 4=1000$

2) When the last two digits have a zero that is $4$ cases

Then the left most digit will have $6$ options since $0$ already used.

The next digit will also have $5$ options

The next place will have only $4$ options since digits cannot be repeated.

Hence the number of possible ways are $=4\times 6\times 5\times 4=480$

Therefore the answer is $1000+480=1480$