In the class where Petya and Vanya study, there are 31 people. In how many ways can a football team be selected from the class?

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The first method. Consider three cases: only Petya is in the team; only Vanya is in the team; both of them are in the team. The second method. How many options are not suitable for us?

Answer

$2 C_{29}^{10}+C_{29}^{11}=C_{31}^{11}-C_{29}^{9}$ ways.