The key to a speedy solution is to realise that all factorials greater than 5! must be divisible by 9 (because 3 and 6 are factors of 6! and 3x6=18, which is a multiple of 9).
Now, it is well known that every number divisible by 9 has its digit-sum also divisible by 9. So we simply need to add up the eighteen digits that have been given, to arrive at 52. Then, the unidentified, leftmost digit must have been 2, to make a grand total - divisible by 9 - of 54.
Hence the full integer is 2,432,902,008,176,640,000
This was first published in May 2009