1. The number of odd coefficient in an expansion

The function considers two cases: when n is odd and when n is even. The naive way.

lb[n_] := Module[{obc={}, ebc={}}, If[EvenQ[n], 

For[i = 0,  i< (n/2)+1, i++,  
If[OddQ[Binomial[n,i]],AppendTo[obc,Binomial[n,i]], AppendTo[ebc,Binomial[n,i]]] 
],
For[k = 0,  k< (n+1)/2, k++,  
If[OddQ[Binomial[n,k]],AppendTo[obc,Binomial[n,k]], AppendTo[ebc,Binomial[n,k]]]
]
]; 
num = {{"Odd:"Length[obc]}, {"Even:"Length[ebc]}} (*Index 1 holds number of distinct odd coefficients, Index 2 holds number of distinct even coefficients*)
]

Another way of finding the odd numbers in the expansion (x + y)^n.  When I understood Tables and the Count functions.

lb1[n_] :=Module[{i},
l = Count[Table[Binomial[n, i], {i,0, n}]//OddQ, True]/2;
Print["The distinct odd number(s) with n = ",n , " is/are:   ", l] 
];

lb1[10000]

?ArrayPlot

ArrayPlot[Table[
Mod[Binomial[n,k],2],
{n,0,100},{k,0,n}]]

?Mod