grandes-ecoles 2020 Q20

grandes-ecoles · France · mines-ponts-maths2__mp_cpge Permutations & Arrangements Lattice Path / Grid Route Counting

We assume that $d = 2$ and that the distribution of $X$ is given by $$P ( X = ( 0,1 ) ) = P ( X = ( 0 , - 1 ) ) = P ( X = ( 1,0 ) ) = P ( X = ( - 1,0 ) ) = \frac { 1 } { 4 }$$ Let $n \in \mathbb{N}$. Establish the equality $$P \left( S _ { 2 n } = 0 _ { 2 } \right) = \left( \frac { \binom { 2 n } { n } } { 4 ^ { n } } \right) ^ { 2 }$$

We assume that $d = 2$ and that the distribution of $X$ is given by
$$P ( X = ( 0,1 ) ) = P ( X = ( 0 , - 1 ) ) = P ( X = ( 1,0 ) ) = P ( X = ( - 1,0 ) ) = \frac { 1 } { 4 }$$
Let $n \in \mathbb{N}$. Establish the equality
$$P \left( S _ { 2 n } = 0 _ { 2 } \right) = \left( \frac { \binom { 2 n } { n } } { 4 ^ { n } } \right) ^ { 2 }$$

Paper Questions

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