IIT-JEE 2007 · Paper 2, Mathematics question 47

Complement rule.
$$P(E^c\cap F^c\mid G)=1-P(E\mid G)-P(F\mid G)+P(E\cap F\mid G).$$

Use the hypotheses. Pairwise independence gives $P(E\mid G)=P(E)$ and $P(F\mid G)=P(F)$, and $P(E\cap F\cap G)=0$ makes the last term zero:
$$P(E^c\cap F^c\mid G)=1-P(E)-P(F)=P(E^c)-P(F).$$

(C)