In an old probability test: $$ P(X_i=1)=n^{-\frac{1}{2}} $$ $$ P(X_i=0)=1-n^{-\frac{1}{2}} $$ $$ S_{i,j,k} = 1 \text{ if }X_i=X_j=X_k=1 \text{ (and 0 otherwise)}$$ $$ Y = \text{Number of }S_{i,j,k}\text{ that are equal to 1}$$
First question was to calculate $\mathbb{E}(Y)$ which I found out to be $\Theta(n^{1.5})$
Second question is to find $Var(Y)$ which i'm not sure how to get done. I know that $Var(Y)=\mathbb{E}(X^2)-\mathbb{E}(X)^2=\mathbb{E}(\mathbb{E}(X-\mathbb{E}(X))^2)$ But I don't know how to solve these equations. I thought that it's just a Binomial distribution with probability of $(n^{-\frac{1}{2}})^3=n^{-\frac{3}{2}}$ which means $Var(Y)=np(1-p)={n\choose{3}}n^{-\frac{3}{2}}(1-n^{-\frac{3}{2}})$.
This gives me $Var(Y)=\Theta(n^{1.5})\text{ }$but according to the answers it should be $\Theta(n^{2.5})$.
Am I wrong because there's covariance/codependency between the different $S_{i,j,k}$ variables? How do I find $Var(Y)$?