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Sum[Sum[5!/(a!b!(5-a-b)!)*(1/2)^a(1/3)^b(1/6)^(5-a-b),{b,0,5-a}],{a,2,5}]
Sum[Sum[5!/(a!b!(5-a-b)!)*(1/2)^a(1/3)^b(1/6)^(5-a-b),{b,0,5-a}],{a,2,5}]
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