Mac[q_]:=Binomial[420,q]*Binomial[180,100-q];

Plot[Mac[q],{q,0,100},PlotTheme->"Scientific",FrameLabel->{"(Subscript[q, A])","(Subscript[Ω, total])"},PlotStyle->Black,PlotRange->All]

Plot[Binomial[420,q],{q,0,100},PlotTheme->"Scientific",FrameLabel->{"(Subscript[q, A])","(Subscript[Ω, A])"},PlotStyle->Black,PlotRange->All]

Plot[Binomial[180,100-q],{q,0,100},PlotTheme->"Scientific",FrameLabel->{"(Subscript[q, A])","(Subscript[Ω, B])"},PlotStyle->Black,PlotRange->All]

Plot[{Log[Binomial[420,q]],Log[Binomial[180,100-q]],Log[Binomial[420,q]*Binomial[180,100-q]]},{q,0,100},PlotTheme->"Scientific",FrameLabel->{"(Subscript[q, A])","S/k"},PlotRange->All,PlotLegends->{"(Subscript[S, A])","(Subscript[S, B])","(Subscript[S, total])"}]

Reverse@{Table[Log[Binomial[420,q]*Binomial[180,100-q]],{q,0,100,1}]//Max,Position[Table[Log[Binomial[420,q]*Binomial[180,100-q]],{q,0,100,1}],Table[Log[Binomial[420,q]*Binomial[180,100-q]],{q,0,100,1}]//Max][[1]][[1]]-1}

Max@Table[Log[Binomial[420,q]*Binomial[180,100-q]],{q,0,100,1}]

Plot[Mac[q]//N,{q,0,100},PlotRange->All]

Mac[100]

Binomial[q+420-1,q]*Binomial[q+180-1,q]

DiscretePlot[Binomial[179+q,q] Binomial[419+q,q],{q,1,100}]

Binomial[q+180-1,q]