Why is the flux 0? I don't understand this concept

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Why does it say that the flux due to q_2 and q_3 through S is 0? Doesn't it contain a nonzero charge q_1?

Does anyone also know the difference between "no charge" vs "net charge is 0"? My book differentiates it.

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The surface $\mathrm{S}$ does indeed contain charge $q_1$, and so will have nonvanishing electric flux. However, $\mathrm{S}$ does not contain charges $q_2$ and $q_3$, so it will have zero total flux due to those charges.