Bucket Sort Expected time

The Book skips small but important math during expansion for the square. Below step is transformed to $$ E[n_i^2] = E\Big[ {\Big( \sum_{j = 1}^n X_{ij} \Big)}^2 \Big] $$ $$ = E\Big[ {\Big( \sum_{j = 1}^n \sum_{k = 1}^n X_{ij} X_{ik} \Big)}^2 \Big] $$ expanding below by expanding summation $$ {\Big( \sum_{j = 1}^n X_{ij} \Big) }^2 $$ $$ \sum_{j = 1}^n X_{ij} . \sum_{j = 1}^n X_{ij} $$ $$ (X_{i1}+X_{i2}+X_{i3}+...+X_{in}) (X_{i1}+X_{i2}+X_{i3}+...+X_{in}) $$ multiplying above

$ X_{i1}. X_{i1}+X_{i1} . X{i2}+....+X_{i1}.X_{in} + $
$ X_{i2}. X_{i1}+X_{i2} . X{i2}+....+X_{i2}.X_{in} + $ . . . $ X_{in}. X_{i1}+X_{in} . X{i2}+....+X_{in}.X_{in} + $

we can group all elements with equal subscripts

$ X_{i1}. X_{i1}+X_{i2} . X{i2}+....+X_{in}.X_{in} + $
$ X_{i1}. X_{i2}+X_{i1} . X{i3}+....+X_{i1}.X_{in} + .... $
The first part can be written as $$ \sum_{j = 1} ^ n X_{ij} $$ The second part can be written as $$ \sum_{j = 1} ^ n \sum_{k = 1} ^ n X_{ij} X_{ik} $$ together we can write as $$ \sum_{j = 1} ^ n X_{ij} + \sum_{j = 1} ^ n \sum_{k = 1} ^ n X_{ij} X_{ik} $$