Question

How many natural numbers are there which are smaller than ${10}^4$and are divisible by $4$, whose decimal notation consists only of the digits $0,1,2,3$ and $5$ which do not repeat in any of these numbers?

Answer 1

There are $5$ ways of choosing $1$ digit $(0.1,2,3,5)$

$4\times 4$ ways of choosing two digit

$4\times 4\times 3$ ways of choosing $3$

$4\times 4\times 3\times 2$ ways of choosing $4$

There are no$:$ $5$ digits no$:$ smaller than ${10}^4$ that start with zero$.$

$\therefore$ Total no$:$ $=5+16+48+96=165$

But the no$:$ must be divisible by $4.$

$\therefore$ Total no$:$ $=0+6+48+96=160.$