$n$ items are distributed into $n$ boxes such that each item is independently put with probability $p_j$ to be put in box $j$, for $j=1,2,\ldots,n$, where $\sum_{j=1}^np_j=1$. A collision occurs whenever an item is put into a non-empty box. Find the expected number of collisions.
What's wrong with the following reasoning? Let $C_i$ be the number of collisions in box $i$. Then $C_i+1$ is the number of items in box $i$, so $E[C_i+1]=np_j$ and $E[C_i]=np_j-1$. Therefore the expected number of collisions is $\sum_{j=1}^n(np_j-1)=0$. Obviously the number cannot be zero, because the number is sometimes positive but never negative.