Question 1210234
That's a colorful sock drawer! Let's figure out the minimum number of socks Vera needs to pull to guarantee six matching pairs of different colors.

To ensure she has six matching pairs of *different* colors, we need to consider the worst-case scenario. Imagine Vera keeps picking one sock of each color before she gets a second one of any color.

Here's the breakdown:

1.  She could pick one white, one black, one brown, one blue, one red, and one green sock. That's 6 socks, and no pairs yet.
2.  She could then pick a second white sock, making one white pair. That's 7 socks.
3.  Next, she could pick a second black sock, making one black pair. That's 8 socks.
4.  She could continue this pattern, picking one of each of the remaining colors to form a pair. After picking a second brown, blue, red, and green sock, she would have 6 pairs of different colors. This would be $6 + 6 = 12$ socks.

However, we need to *ensure* at least six matching pairs of *different* colors. Consider the absolute worst luck:

* She could pick all 20 white socks.
* Then, she could pick all 21 black socks.
* Then, she could pick all 22 brown socks.
* Then, she could pick all 23 blue socks.
* Then, she could pick all 24 red socks.

At this point, she has $20 + 21 + 22 + 23 + 24 = 110$ socks, and she still doesn't have six matching pairs of *different* colors. The very next sock she picks *must* be green, creating her first green pair.

Now, to guarantee six different colored pairs, after picking all of the socks of five colors, she would need to pick two socks of the sixth color.

So, the worst-case scenario to guarantee six matching pairs of different colors is:

* Pick all the socks of 5 colors (the largest quantities): $25 + 24 + 23 + 22 + 21 = 115$ socks.
* Then, pick 2 socks of the remaining color (white) to form a sixth pair.

Therefore, Vera must pull out a minimum of $115 + 2 = 117$ socks to ensure at least six matching pairs of different colors.