I am not sure whether this is the best or easiest way but here is one way to attack the problem. I will use L to represent an arbitrary letter, 9 for an arbitrary digit, and * for a special character.
Case 1 – Exactly one number
Case 1a – The letter is in position 7 and the special character is in 8 i.e. LLLLLL9* so, there are 26^6 x 10 x 9 possibilities.
Back to the general case 1, how many other arrangements are there? The digit can be in any of 8 positions and the special character in any of the other 7. So 8 x 7 x the number of 1a cases.
Case 2 – Exactly two numbers
Case 2a – the pattern LLLLL99* has 26 ^ 5 x 10 ^ 2 x 9 possibilities.
Back to general case 2, how many other arrangements? The two digits can be in 8 x 7 / 2 places and the special character in one of the remaining 6.
Case 3 etc. Continue in the same way. You will need some simple combinatorics e.g. how many arrangements of three digits, four digits, etc.
The rule appears to allow case 7: 0 letters, 7 digits and 1 special character so don’t forget that.