Cardinality constraint (${\sum }_i{X}_i=k,X_i\in \{0,1\}$)

A cardinality constraint restricts how many members of a specified collection are selected or satisfy a property. For Boolean variables, exactly $k$ selected entries gives the displayed sum; upper and lower bounds give at-most and at-least variants. Constraint propagation can maintain the number already selected and the number still available. A requirement involving many variables is a global constraint, even when it has a compact arithmetic description.