-
$\displaystyle\binom{4}{2}$;
Corrigé
$\displaystyle\binom{4}{2}=6$.
\[\begin{array}{|c|c|c|c|c|c|c|c|} \hline
\scriptstyle{n\downarrow,k\rightarrow}&\scriptstyle{0} &\scriptstyle{1}&\scriptstyle{2}&\scriptstyle{3}
&\scriptstyle{4}&\scriptstyle{5}&\scriptstyle{6}\\ \hline
{\scriptstyle0} & 1 & & & & & & \\ \hline
{\scriptstyle1} & 1 & 1 & & & & & \\ \hline
{\scriptstyle2} & 1 & 2 & 1 & & & & \\ \hline
{\scriptstyle3} & 1 & 3 & 3 & 1 & & & \\ \hline
{\scriptstyle4} & 1 & 4 &{\color{Red}6}& 4 & 1 & & \\ \hline
{\scriptstyle5}& 1 & 5 & 10 & 10 & 5 & 1 &\\ \hline
{\scriptstyle6}& 1 & 6 & 15 & 20 & 15 & 6 & 1 \\ \hline
\end{array}\]
-
$\displaystyle\binom{5}{2}$;
Corrigé
$\displaystyle\binom{5}{2} = 10$.
\[\begin{array}{|c|c|c|c|c|c|c|c|} \hline
\scriptstyle{n\downarrow,k\rightarrow}&\scriptstyle{0} &\scriptstyle{1}&\scriptstyle{2}&\scriptstyle{3}
&\scriptstyle{4}&\scriptstyle{5}&\scriptstyle{6}\\ \hline
{\scriptstyle0} & 1 & & & & & & \\ \hline
{\scriptstyle1} & 1 & 1 & & & & & \\ \hline
{\scriptstyle2} & 1 & 2 & 1 & & & & \\ \hline
{\scriptstyle3} & 1 & 3 & 3 & 1 & & & \\ \hline
{\scriptstyle4} & 1 & 4 & 6 & 4 & 1 & & \\ \hline
{\scriptstyle5}& 1 & 5 &{\color{Red}10}& 10 & 5 & 1 &\\ \hline
{\scriptstyle6}& 1 & 6 & 15 & 20 & 15 & 6 & 1 \\ \hline
\end{array}\]
-
$\displaystyle\binom{6}{3}$;
Corrigé
$\displaystyle\binom{6}{3} = 20$.
\[\begin{array}{|c|c|c|c|c|c|c|c|} \hline
\scriptstyle{n\downarrow,k\rightarrow}&\scriptstyle{0} &\scriptstyle{1}&\scriptstyle{2}&\scriptstyle{3}
&\scriptstyle{4}&\scriptstyle{5}&\scriptstyle{6}\\ \hline
{\scriptstyle0} & 1 & & & & & & \\ \hline
{\scriptstyle1} & 1 & 1 & & & & & \\ \hline
{\scriptstyle2} & 1 & 2 & 1 & & & & \\ \hline
{\scriptstyle3} & 1 & 3 & 3 & 1 & & & \\ \hline
{\scriptstyle4} & 1 & 4 & 6 & 4 & 1 & & \\ \hline
{\scriptstyle5}& 1 & 5 & 10 & 10 & 5 & 1 &\\ \hline
{\scriptstyle6}& 1 & 6 & 15 &{\color{Red}20}& 15 & 6 & 1 \\ \hline
\end{array}\]
-
$\displaystyle\binom{6}{4}$.
Corrigé
$\displaystyle\binom{6}{4} = 15$.
\[\begin{array}{|c|c|c|c|c|c|c|c|} \hline
\scriptstyle{n\downarrow,k\rightarrow}&\scriptstyle{0} &\scriptstyle{1}&\scriptstyle{2}&\scriptstyle{3}
&\scriptstyle{4}&\scriptstyle{5}&\scriptstyle{6}\\ \hline
{\scriptstyle0} & 1 & & & & & & \\ \hline
{\scriptstyle1} & 1 & 1 & & & & & \\ \hline
{\scriptstyle2} & 1 & 2 & 1 & & & & \\ \hline
{\scriptstyle3} & 1 & 3 & 3 & 1 & & & \\ \hline
{\scriptstyle4} & 1 & 4 & 6 & 4 & 1 & & \\ \hline
{\scriptstyle5}& 1 & 5 & 10 & 10 & 5 & 1 &\\ \hline
{\scriptstyle6}& 1 & 6 & 15 & 20 &{\color{Red}15}& 6 & 1 \\ \hline
\end{array}\]