To determine the minimum number of balls needed to ensure that the drawn balls contain at least one full group of any color, we find the maximum number of balls we can draw without selecting a complete group of any color. By drawing one less ball from each color, we have a total of 174 balls. Adding just one more ball, regardless of its color, completes a group. Hence, the minimum number of balls required, n, is

174+1=175

For ensuring that the drawn balls include at least one ball of each color, we aim to maximize the number of balls drawn without selecting any black balls, as black balls are the least in number. We can draw 20 blue balls, 26 green balls, 28 yellow balls, 38 red balls, and 54 white balls, totaling 166 balls without any black balls. Adding just one black ball completes the requirement, making the minimum number of balls needed

166+1=167