Degrees of freedom for testing homogeneity of $r$ multinomial populations

To test the equality of $r$ independent multinomial distributions, multinomial$(k,p_{i_1},\dots,p_{i_k}) i=1\dots r$ the null hypothesis is $H_0:P_{ij}=P_j \forall j=1\dots k$ where $p_j$ are unknown $j=1\dots k$, a $\chi^2$ test statistic is used. The degrees of freedom of the $\chi^2$ test are: