Number of semi-standard Young tableaux of shape $\lambda$ with some entries fixed

Question

Given a partition $\lambda$, the number of semi-standard Young tableaux (SSYT) of shape $\lambda$ with maximum entry $n$ is given by \begin{equation} \prod_{1\leq i<j\leq n} \frac{\lambda_i-\lambda_j-i+j}{j-i}. \end{equation} Instead of counting all the tableaux, can something be said when we impose the condition that certain entries in the tableaux are fixed? For example, the number of SSYT of shape $(2,2)$ containing ${1,2}$ with maximum entry 3 is 4. How do we find the number of SSYT for arbitrary $\lambda$ and fixed set of entries ${i_1,\dots, i_k}$ with maximum entry $n$?