Question 1210234
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Vera has 20 white socks, 21 black socks, 22 brown socks, 23 blue socks, 24 red socks, and 25 green socks. 
How many socks (at a minimum) must she pull out of her sock drawer to ensure at least six matching pairs 
of different colors?
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In the worst case, Vera will take 25 green socks, 24 red socks, 23 blue socks,
22 brown socks and 21 black socks - in all, 25 + 24 + 23 + 22 + 21 = 115 socks.



           Notice that here I list the socks in the reverse order - 
           according to their numbers of each color in the DESCENDING order.



Then she will have a lot of matching pairs of 5 colors (green, red, blue, brown and black), 
but still will NOT have 6 matching pairs of different color.


But after that, any two next socks will create the required 6-th pair of the white color.


So, the minimum number of socks Vera should draw from the drawer is 115 + 2 = 117,
to have 6 guaranteed matching pairs of 6 different colors.



My answer coincides with the answer by @CPhill, but explanation/reasoning/logic
is slightly different (more straightforward, in order for do not confuse a reader).



Solved.