Combination of $4$ digit numbers not divisible by $5$

Question

The number of $4$ digit numbers which are not divisible by $5$ that can be formed using the digits $(0,2,4,5)$ if digits are not repeated is?

Answer 1

Well:

• It cannot start with $0$ (because it must be a $4$-digit number)
• It cannot end with $0$ or $5$ (because it must not be divisible by $5$)

This leaves you with only $8$ options:

• $2054$
• $2504$
• $4052$
• $4502$
• $5024$
• $5042$
• $5204$
• $5402$

Answer 2

There are 12 such numbers

$0254$$\hspace{0.9cm}$ $0524$$\hspace{0.9cm}$ $2054$$\hspace{0.9cm}$ $2504$$\hspace{0.9cm}$

$5024$$\hspace{0.9cm}$ $5204$$\hspace{0.9cm}$ $0452$$\hspace{0.9cm}$ $0542$

$4052$$\hspace{0.9cm}$ $4502$$\hspace{0.9cm}$ $5042$$\hspace{0.9cm}$ $5402$

4 are 3 digit numbers$\hspace{0.9cm}$ $0254$$\hspace{0.9cm}$$0524$$\hspace{0.9cm}$$0452$$\hspace{0.9cm}$$0542$

So the answer is 8

Answer 3

$$4!-(3!+3!+3!)+2!=8$$ $4!$ for all permutations,

$3!$ for the numbers starting with $0$, ending on $0$ and ending on $5$ respectively,

$2!$ for the numbers starting with $0$ and ending on $5$.