SOLUTION: Toni has 10 identical blue socks, 14 identical green socks, and 4 identical red socks in a drawer. she removes socks from the drawer one at a time at random without looking. The le

Question 1198918: Toni has 10 identical blue socks, 14 identical green socks, and 4 identical red socks in a drawer. she removes socks from the drawer one at a time at random without looking. The least number of socks that Toni needs to pull out to be sure of having 4 pairs of the same colour is
a) 8 b) 15 c) 16 d) 19 e) 22
Found 2 solutions by math_tutor2020, greenestamps:
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

B = blue
G = green
R = red

4 pairs = 4*2 = 8 individual socks
Of course it's impossible to have 4 pairs of red, since Toni simply doesn't have enough red socks.
Therefore, she'll either have 8 blue or 8 green.

The best case scenario is that Toni selects 8 blue socks in a row, or 8 green socks in a row.
This would be the minimum number of selections needed.
This is if she gets really lucky.

But we cannot assume this incredible luck and instead have to consider the worst case scenario, so that we 100% guarantee getting those 4 matching pairs.

The worst case scenario is her selecting say a blue sock first, then green, then red.
It would give the sequence: B, G, R.
The order doesn't really matter.
All that matters is that she doesn't have a matching pair yet.

The 4th selection changes that since she has no other colors to pick from.
This 4th selection is guaranteed to be a match with a previous selection.
Refer to the pigeonhole principle.

Let's say the 4th selection was red.
Then let's say the 5th and 6th selections were also red.
This means she has chosen all 4 red socks.

But as I mentioned earlier, we ignore red since there aren't enough red socks to make 4 pairs that match.

We have this sequence so far
B, G, R
R, R, R

For the 7th selection we could go for blue or green.
Let's have the colors alternate to guarantee the worst case scenario.
This delays a matched pair as much as possible.

She has selected 1 blue sock so far.
There are 10-1 = 9 blue socks left.
She needs 8-1 = 7 more blue socks to have 4 pairs of blue.

She has selected 1 green sock so far.
There are 14-1 = 13 green socks left.
She needs 8-1 = 7 more green socks to have 4 pairs of green.

In short: she needs 7 more of either green or blue.

The worst case scenario is that she selects 6 green and 6 blue in any order you want.
That's an additional 6*2 = 12 socks on top of the 6 already chosen.
There are a total of 6+12 = 18 socks selected so far.

By the 19^th selection, she has 4 pairs of the same color (aka 8 socks of the same color).
We don't have enough information to determine if the 4 matching pairs would be blue or green.

Answer: 19 (choice D)

Bonus question to think about:
What is the least number of socks that Toni needs to select to guarantee 4 pairs of blue and 4 pairs of green?

Edit: The tutor @greenestamps offers a more streamlined approach. And it makes sense to get the red socks out of the way first, which I didn't consider.

Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

You need to consider the worst case scenario on a problem like this. In this problem we want to find the largest number of socks we can pull out of the drawer WITHOUT getting 4 pairs of the same color.

There are only 4 red socks, so we will never get 4 pairs of red socks. So in our worst case scenario we pick all 4 of the red socks.

There are enough blue and green socks to make four pairs of either color. To get 4 pairs of either color, we need 8 socks of that color. So our worst case scenario is to get 7 blue and 7 green before we get 8 of either color.

So in the worst case, without getting 4 pairs of one color, we can get 4 red, 7 blue, and 7 green, for a total of 18 socks.

After that, the 19th sock we pull out of the drawer must be either blue or green, giving us 4 pairs of that color.

ANSWER: Toni needs to pull 19 socks out of the drawer to be sure she has 4 pairs of the same color.

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