A conducting sphere of radius R carrying charge Q is connected to an uncharged conducting sphere of radius 2R .The charge that flows between them is

We know that charge will flow between them till the potential becomes same on both spheres.

Let us assume that a charge x flows between them

charge on larger sphere will be x and charge on smaller sphere will be Q - x

Potential on smaller sphere = k (Q - x) / R

Potential on larger sphere = k x /2R

Equating the potentials

k (Q - x) / R = k x /2R

Q - x = x/2

3x/2 = Q

x = 2Q/3

Thus a charge of 2Q/3 will flow between them

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However if you consider them concentric spheres, such that smaller sphere lies inside the larger sphere. Then potential on inner sphere will be always greater. If you connect them, the whole charge Q will be emptied from the inner sphere