Answer
$\displaystyle \frac{7}{9}$
Work Step by Step
$P(E)=\displaystyle \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}=\frac{n(E)}{n(S)}$
$n(S)=C(10,5)=252$
$E$ = no more than 1 white = (exactly $1$ white) OR (exactly $0$ whites)
n (exactly $1$ white) = n(exactly $1$ white AND $4$ of the other 8)=
$=C(2,1)\cdot C(8,4)=2\cdot 70=140$
n(exactly $0$ whites)=n(exactly $0$ white AND $5$ of the other 8)
$=C(2,0)\cdot C(8,5)=1\cdot 56=56$
$n(E)=140+56=196$
$P(E)=\displaystyle \frac{196}{252}=\frac{7}{9}$