With the notation of question 28, justify that, for all $j \in \llbracket 1 , n - 1 \rrbracket$,
$$\mathbb { P } \left( C _ { j + 1 } \in \operatorname { Vect } \left( C _ { 1 } , \ldots , C _ { j } \right) \right) = \sum _ { \left( v _ { 1 } , \ldots , v _ { j } \right) \in \mathcal { V } _ { n , 1 } ^ { j } } \mathbb { P } \left( C _ { j + 1 } \in \operatorname { Vect } \left( v _ { 1 } , \ldots , v _ { j } \right) \right) \mathbb { P } \left( \left( C _ { 1 } = v _ { 1 } \right) \cap \cdots \cap \left( C _ { j } = v _ { j } \right) \right) .$$