7. List all six-digit numbers that can be written using the digits $1,2,3,4$, 5 and 6, with each digit appearing exactly once. How many of these numbers are greater than 345,612, and how many are less than that number?

No use of a pocket calculator or any manuals is allowed.
First method:

Let $\overline{a b c d e f}$ be such that $a, b, c, d, e, f \in\{1,2,3,4,5,6\}$ and $a, b, c, d, e, f$ are all distinct.

There are a total of $6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=720$ numbers.

2 POINTS Of these 720, the numbers less than 345612 are:

- $\overline{1 b c d e f}, \overline{2 b c d e f}$

There are $5 \cdot 4 \cdot 3 \cdot 2 \cdot 1+5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=2 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=240$ such numbers. 2 POINTS

- $\overline{31 c d e f}, \overline{32 c d e f}$

There are $4 \cdot 3 \cdot 2 \cdot 1+4 \cdot 3 \cdot 2 \cdot 1=2 \cdot 4 \cdot 3 \cdot 2 \cdot 1=48$ such numbers. 2 POINTS

- $\overline{341 d e f}, \overline{342 d e f}$

There are $3 \cdot 2 \cdot 1+3 \cdot 2 \cdot 1=2 \cdot 3 \cdot 2 \cdot 1=12$ such numbers. 1 POINT

- $\overline{3451 e f}, \overline{3452 e f}$

There are $2 \cdot 1+2 \cdot 1=2 \cdot 2 \cdot 1=4$ such numbers. 1 POINT

The total number of numbers less than 345612 is $240+48+12+4=304$.

1 POINT

The number of numbers greater than 345612 is $720-304-1=415$.

1 POINT

TOTAL 10 POINTS