10. If real numbers $a, b, c, d, e$ satisfy $a+b+c+d+e=8, a^{2}+b^{2}+c^{2}+d^{2}+e^{2}=16$, then the sum of the maximum and minimum values of $e$ is $\qquad$
10. To find the maximum and minimum values of $e$, isolate $e$ in the given conditions, i.e., transform the known equations to:
$a+b+c+d=8-e, a^{2}+b^{2}+c^{2}+d^{2}=16-e^{2}$, construct a quadratic function
$$
f(x)=4 x^{2}-2(a+b+c+d) x+a^{2}+b^{2}+c^{2}+d^{2}=(x-a)^{2}+(x-b)^{2}+(x-c)^{2}+(x-d)^{2} \text {. }
$$

Then $f(x) \geqslant 0$ for all $x \in \mathbf{R}$.
Since the coefficient of the quadratic term in $f(x)$ is greater than 0,
we have $\Delta=4(8-e)^{2}-16\left(16-e^{2}\right) \leqslant 0$, i.e., $5 e^{2}-16 e \leqslant 0 \Rightarrow 0 \leqslant e \leqslant \frac{16}{5}$.
Therefore, the maximum and minimum values of $e$ are $\frac{16}{5}$ and 0, respectively, and their sum is $\frac{16}{5}$.