[tex]\displaystyle\\
\text{Pentru }~~C_n^k ~~\text{ avem urmatoarele conditii pentru }~n\text{ si }k:\\\\
C1)~~~n,~k \in N\\
C2)~~~n\geq 1\\
C3)~~~0\leq k \leq n\\\\
\Longrightarrow~~\log_2 x~~si ~~\log_4x \in N\\\\
\Longrightarrow~~x~~\text{este o putere a lui 4},~x \neq 0
[/tex]
[tex]\displaystyle\\
x=4^0 = 1\\
\log_2 1 = 0\\
2+\log_4 1 = 2+0=2\\
C_2^0=\boxed{1}\\\\
x=4^1 = 4\\
\log_2 4 = 2\\
2+\log_4 4 = 2+1=3\\
C_3^2=\frac{A_3^2}{P_2}=\frac{3\cdot 2}{1\cdot 2}=\boxed{3}\\\\
x=4^2=16\\
\log_2 16 = 4\\
2+\log_4 16 = 2+2=4\\
C_4^4 = \frac{A_4^4}{P_4}= \frac{4\cdot3\cdot2\cdot1}{1\cdot 2\cdot3\cdot4}=\boxed{1}\\\\
x=4^3=64\\
\log_2 64 = 6\\
2+\log_4 64 = 2+3=5\\
6 \ \textgreater \ 5~~~\text{Nu respecta conditia: }~~ 0 \leq k \leq n
[/tex]
[tex]\displaystyle\\
x\in \{1;~4;~16\}\\\\
C_{2+\log_4x}^{\log_2 x} = \begin{cases}
1~~\text{daca}~~x=1\\
3~~\text{daca}~~x=4\\
1~~\text{daca}~~x=16
\end{cases}\\\\\\
\text{Alte valori pentru }~x~\text{ nu sunt admise.} [/tex]
