What is the number of strings of length $235$ which can be made from the letters A, B, and C, such that the number of A's is always odd, the number of B's is greater than $10$ and less than $45$ and the Number of C's is always even?
What I can think of is
$\left(\binom{235}{235} - \left\lfloor235 - \frac{235}2\right\rfloor\right) \binom{235}{35} \binom {235}{ \lfloor 235/2\rfloor}\;.$
Thanks