Maximum number of possible keys for a fixed-length password?

I'm reviewing some information and possibilities regarding key space size following a unique password requirement, as it were.

This particular requirement is a fixed-length password of 12 characters, with exactly 1 capital letter, and exactly 1 digit.

My understanding is that, let's say we have 26 letters, then we'd take 26^{(number of characters)} so for a 10 character password, your possible key combos is in the neighborhood of 141,167,095,653,376.

If you included all upper case and lower case, that'd be 52^{(number of characters)}, so on and so forth--the question I'm stumped on is how that changes if you have exactly 1 of a UC and exactly 1 of a digit, if it does at all.

Key space = $26 \times 26^{10} \times 10 = 3.67 \times 10^{16}$ for a capital letter followed by ten lower case letters and one number. The order doesn't matter, and I assume that your input routine enforces the upper and lower case format you've chosen.
A password space with $n$ lower case letters, 1 upper case letters and one digit in any order is of size: $26^{n+1}*10*(n+2)*(n+1)$