How many different ways can you arrange the numbers 1 to 9?

If all the numbers have to be used each time then there are 9 ways of choosing the first number. After that there are 8 ways of choosing the second number. This presumes that we can't use the same number in two different places in the one arrangement. Say we choose 4 as the first number, the second could be any of 1,2,3,5,6,7,8 or 9 Already we have 72 permutations and we've only got the first two numbers sorted. We not have 7 numbers left from which to choose the third, and so on. To go all the way down to the last number we end up with 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 permutations. Work it out, it's a big number.