I read the book of Introduction to Probability SECOND EDITION by John N. Tsitsiklis et al. and in Sec. 4.3 Conditional Expectation and Variance Revisited, I think the $var(X|Y)$ is not $nY(1-Y)$ but $nY(1-nY)$.
My understanding is that since $var(X|Y)=E(X^2|Y)-[E[X|Y]]^2$ and $E[X|Y]=nY$ and $E(X^2|Y)=nY$, so $var(X|Y)=nY(1-nY)$ rather than $nY(1-Y)$.
How do you think about it?