I read this as: what is the minimum number of balls that must be drawn to guarantee at least two balls of each colour.

If so:

If he draws 15 balls, by chance he could pick all the yellow balls but no blue or green. So he must draw more than 15.

If he draws 25 balls, by chance he could pick all the yellow and blue balls but no green. So he must draw more than 25.

To ensure at least two greens, he should draw 27. All 28 possible outcomes would be:

(yellow,blue,green)=

(15,10,2), (15,9,3), ..., (15,4,8),

(14,10,3), (14,9,4), ..., (14,5,8),

(13,10,4), (13,9,5), ..., (13,6,8),

(12,10,5), (12,9,6), (12,8,7), (12,7,8),

(11,10,6), (11,9,7), (11,8,8),

(10,10,7), (10,9,8),

(9,10,8).

So he would draw a minimum of 2 greens, 4 blues and 9 yellows.