of the Backstroke, Breaststroles four different swimmers, who successively swim $100$ meters whose expected times $($in second$)$ Butterfly, and Freestyle. A coach has $4$ very fast swimmers coach wants to assign the swimmers in the individual events are given in the following table. The

Event

$\backslash$table$[[$Swimmer$,$Breaststroke,Backstroke,Butterfly,Freestyle$],[$Alison$,80,65,60,55],[$Beth$,65,57,65,52],[$Carla$,75,55,63,50],[$Denise$,70,75,58,54]]$

Let Breaststroke be represented by $1,$ Backstroke by $2,$ Butterfly by $3$ and Freestyle by $4$ and the swimmers be referred to by the first alphabet of their names i$.$e$.$ Alison is A etc. Let $x_{ij}$ represent swimmer $"$i$")($A$,$ B$,$ C or D$)$ assigned to event $j"($ or $4$