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

Question

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.

Answer 1

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 $12345= 27216-8\cdot7\cdot6-7\cdot6-6-2=26830$

Answer 2

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, $1234e,\,e\gt5$.