Given,

Bag have $6$ white and $4$ black balls

Now also given die is rolled and the number denotes the number of ball drawn from the bag,

So, probability all drawn balls are white will be,

$=\frac{1}{6}\left[\frac{{}^{6}C_1}{{}^{10}C_1}+\frac{{}^{6}C_2}{{}^{10}C_2}+\frac{{}^{6}C_3}{{}^{10}C_3}+\frac{{}^{6}C_4}{{}^{10}C_4}+\frac{{}^{6}C_5}{{}^{10}C_5}+\frac{{}^{6}C_6}{{}^{10}C_6}\right]$

$=\frac{1}{6}\left[\frac{6}{10}+\frac{15}{45}+\frac{20}{120}+\frac{15}{210}+\frac{6}{252}+\frac{1}{210}\right]$

$=\frac{504}{2520}=\frac{1}{5}$