Combinatorics

Say we have a string of the form AABCCC where A is a letter, C is a digit, B can be either and there are no repeated letters or digits.

Can somebody explain why the number of permutations is equal to:

26 * 25 * (24 + 7) * 10 * 9 * 8

and not:

26 * 25 * (24 + 10) * 9 * 8 * 7

Break up the ()s.
The 1st = (26*25*[24]*10*9*8) + (26*25*[7]*10*9*8)
The 2nd = (26*25*[24]*9*8*7) + (26*25*[10]*9*8*7)

The right terms are equal, and correctly account for the case where B is a #.

The left terms are not equal, and the wrong one incorrectly restricts the Cs, since now B is a letter.