Math Is Fun Forum

karney

Guest

combinations problem

A drawer contains red socks, black socks, and white socks.  what is the least number of socks that must be taken out of the drawer to be sure of having 7 pairs of matching socks?   Please explain.  Thanks!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: combinations problem

Hi karney;

If you are drawing them out one at a time then you need 14 socks to have 7 pairs. If you have 11 socks you cannot have 7 pairs of matching socks. So the minimum amount you have to pull out of the drawer is 14.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: combinations problem

I interpret this question differently, as asking how many socks is always enough.

If you pull out 14 socks, then you might get 7 red, 4 black and 3 white. In this case, you can get 3 red pairs, 2 black and 1 white, and have two odd socks left. But then you haven't managed to make 7 pairs.

In fact, if another black sock was in there then you still couldn't get another pair so 15 socks isn't always enough either.

16 socks is fine though.
If you have a group of 4 socks, then it must be possible to find a pair in them because there are only 3 different colours.

16 is more than 4, so a group of 16 socks will have a pair.
Now we have one pair and 14 socks. But by similar logic, this can be turned into 2 pairs and 12 socks.

Continue like this and we eventually end up at 6 pairs and 4 socks. But we know that 4 socks must contain a pair, so the last pair we need can be found.

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Why did the vector cross the road?
It wanted to be normal.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: combinations problem

Hi mathsyperson;

  I agree, because chapters on  the Dirichlet drawer principle (pigeonhole principle) usually demonstrate it with a problem and solution like you are describing. I didn't know if that was what he was after so I went with the simplest ( I thought) answer, figuring karney would let me know if it wasn't what he wanted.

Last edited by bobbym (2009-05-28 20:26:51)

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.