SOLUTION: 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 ensu

Question 1210234: 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?

Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source): You can put this solution on YOUR website!
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.
Answer by ikleyn(52776)   (Show Source): You can put this solution on YOUR website!
.
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?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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.

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