Odd-only subsets with two of 1, 2, 3

Consider subsets of $\{1,2,\dots,10\}$ in which every element is odd and which contain exactly two of the numbers $1,2,3$. How many such subsets are there?

$8$. The only odd pair from $\{1,2,3\}$ is $\{1,3\}$, so both must be included. The other odd numbers $5,7,9$ may each be in or out freely, giving $2^3=8$ subsets.