grandes-ecoles 2011 QIV.B.1

grandes-ecoles · France · centrale-maths2__mp Not Maths

For $n \in \mathbb{N}$ and $k \in \llbracket 0; n \rrbracket$, we denote by $\binom{n}{k}$ the binomial coefficient $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.
Let $p \in \mathbb{N}^*$. Show that $\binom{2p}{p}$ is an even integer.
Deduce that, if $n \in \mathbb{N}^*$ and $p \in \llbracket 1; n \rrbracket$, then $\binom{n+p}{p}\binom{n}{p}$ is an even integer.

For $n \in \mathbb{N}$ and $k \in \llbracket 0; n \rrbracket$, we denote by $\binom{n}{k}$ the binomial coefficient $\binom{n}{k} = \frac{n!}{k!(n-k)!}$.

Let $p \in \mathbb{N}^*$. Show that $\binom{2p}{p}$ is an even integer.

Deduce that, if $n \in \mathbb{N}^*$ and $p \in \llbracket 1; n \rrbracket$, then $\binom{n+p}{p}\binom{n}{p}$ is an even integer.

Paper Questions

QI.A.1 QI.A.2 QI.B.1 QI.B.2 QI.B.3 QI.C QI.D QII.A QII.B QII.C QII.D QII.E QII.F QIII.A.1 QIII.A.2 QIII.A.3 QIII.A.4 QIII.A.5 QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIII.B.5 QIV.A.1 QIV.A.2 QIV.A.3 QIV.A.4 QIV.A.5 QIV.A.6 QIV.B.1 QIV.B.2 QIV.B.3