Cevap: $\boxed D$
$6\geq x\geq y\geq z\geq w\geq v\geq 0$ olsun. $5x^2\geq 40 \Rightarrow x^2\geq 8 \Rightarrow 6\geq x\geq 3$ olur.
$i) x=3$ ise,
$$4y^2\geq y^2+z^2+w^2+v^2=31 \Rightarrow 3=x \geq y \geq 3 \Rightarrow y=3$$ $$3z^2 \geq z^2+w^2+v^2=22 \Rightarrow z\geq 3 \Rightarrow z=3$$ $$w^2+v^2=13 \Rightarrow (w,v)=(3,2) \Rightarrow (x,y,z,w,v)=(3,3,3,3,2)$$ bulunur.
$ii) x=4$ ise,
$$4y^2\geq y^2+z^2+w^2+v^2=24 \Rightarrow 4\geq y\geq 3$$ $y=3$ ise,
$$3z^2\geq z^2+w^2+v^2=15 \Rightarrow z=3$$ $$w^2+v^2=6$$ Buradan çözüm gelmez. $y=4$ ise,
$$z^2+w^2+v^2=8 \Rightarrow (z,w,v)=(2,2,0) \Rightarrow (x,y,z,w,v)=(4,4,2,2,0)$$ bulunur.
$iii) x=5$ ise,
$$4y^2\geq y^2+z^2+w^2+v^2=15 \Rightarrow 3\geq y\geq 2$$ $y=2$ ise,
$$3z^2\geq z^2+w^2+v^2=11 \Rightarrow z=2$$ $$w^2+v^2=7$$ Buradan çözüm gelmez. $y=3$ ise,
$$z^2+w^2+v^2=6 \Rightarrow (z,w,v)=(2,1,1) \Rightarrow (x,y,z,w,v)=(5,3,2,1,1)$$ bulunur.
$iv) x=6$ ise
$$y^2+z^2+w^2+v^2=4\Rightarrow (x,y,z,w,v)=(6,2,0,0,0),(6,1,1,1,1)$$ çözümleri bulunur. Permütasyonlarını alırsak $\dfrac{5!}{4!}+\dfrac{5!}{2!\cdot 2!}+\dfrac{5!}{2!}+\dfrac{5!}{4!}+\dfrac{5!}{3!}=120$ bulunur.
