Is it true that when you mix two volumes of identical gas, there is no entropy change?

Hi, Everyone!

My textbook says that if two volumes of identical gas are mixed, there
is no change in entropy. Suppose I have vessel one (V1) and vessel 2
(V2). V1 and V2 are separated by a removable separator. V1 has 9
positions for molecules and has 2 molecules. V2 has 9 positions for
molecules and has 2 molecules. The number of molecular arrangements in
V1 is (9 choose 2) = 36. The same is true for V2. Thus my system, which
is composed of V1 and V2 has 36 * 36 = 1296 possible states. Suppose
the separator between V1 and V2 is removed. Gases diffuse. We now have
9+9 = 18 positions for molecules and 2+2 = 4 molecules in our system.
(18 choose 4) = 3060. Thus the entropy has increased. How come?

Thank You Very Much.

Truly Yours, Simon Dexter

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