# How many 5 digit numbers can be made by using 0~9, that is bigger than 12345?

So using numbers from 0~9, making a 5 digit number, how many numbers can be formed that is bigger than 12345?

Repetition is not allowed.

Thank you.

• What have you tried so far? You may want to try thinking about how you can pick on digit at a time: if you pick say $2$ as the first digit, how many numbers can you create? If you pick $1$ as the first, what can you pick for the second? – Brian Apr 27 at 23:39
• I know that If I pick 1 as the first, I can pick 2~9 for the second, but it doesn't mean that I can pick 3~9 for the third. This is because If I picked 3 for the second, I can pick 0 for the third. (for ex 13082). – Sadpersonn Apr 27 at 23:53
• 9x9x8x7x6= 27216 and 1x2x2x2x2= 16, so 27216-16=27200. Is this correct? – Sadpersonn Apr 28 at 0:02

Finding the number of $$5$$ digit numbers without repetition and how many of said numbers $$\leq 12345$$ would gives us the number of numbers without repetition $$> 12345$$.

Number of $$5$$ digit numbers without repetition = $$9\cdot9\cdot8\cdot7\cdot6 = 27216$$.

The number of $$5$$ digit numbers without repetition $$\leq12345$$ has 4 cases:

1. The number of $$5$$ digit numbers without repetition of the form $$10$$_ _ _ $$= 8\cdot7\cdot6$$
2. The number of $$5$$ digit numbers without repetition of the form $$120$$ _ _$$=7\cdot6$$
3. The number of $$5$$ digit numbers without repetition of the form $$1230$$ _$$=6$$
4. The number of $$5$$ digit numbers without repetition of the form $$1234$$ _$$=2$$

Number of $$5$$ digit numbers without repetition greater than $$12345= 27216-8\cdot7\cdot6-7\cdot6-6-2=26830$$

• Thank you very much for the answer. I now understood. – Sadpersonn Apr 28 at 0:16

Or, to do it directly, we have $$8\cdot 9\cdot 8\cdot7\cdot6+7\cdot 8\cdot 7\cdot 6+6\cdot 7\cdot 6+5\cdot 6+4=24192+2352+252+30+4=26830$$.

Here I have just added the number of five digit numbers without repitition greater than $$12345$$ of the form $$abcde,\, a\gt1$$, $$1bcde,\,b\gt2$$, $$12cde,\,c\gt3$$, $$123de,\,d\gt4$$ and, finally, $$1234e,\,e\gt5$$.