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Friday, April 22, 2011

anonymous and Allison on choosing 2 out of 6 vs choosing 4 out of 6

I mentioned in a comments thread the fact that I'm having a very difficult time grokking the idea that the number of ways you can choose 2 items out of 6 is the same as the number of ways you can choose  4 items out of 6.

Why does 6C2 = 6C4?

Here is Allison's explanation of the specific problem I was asking about.

And here is Anonymous explaining why 6C2 = 6C4:
Image you have six balls (red, orange, yellow, green, blue, and purple, for example). You also have a box.

6C2: I have six balls, and I am choosing which two balls to put in the box.

6C4: I have six balls, and I am choosing which four to leave out of the box.

Thus, the combination "I put red and orange in the box," is the same way of splitting up the balls as "I leave yellow, green, blue, and purple out of the box." Counting the ways to put two in the box is the same as counting the ways to leave four out of the box. 
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5 comments:

  1. AnonymousApril 22, 2011 at 7:41 PM

    I hope this explanation isn't overkill, so here goes:

    Consider the following fun activity:
    (1) Start with 6 balls, each engraved with a unique ID (like a social-security number).
    (2) Get a supply of blank labels. Write "IN-THE-BOX" on 4 labels and "OUTSIDE-THE-BOX" on 2 labels.
    (3) Place the 6 labels on the 6 balls any old way.


    Now suppose that you and I both do this same fun activity. (We each use 6 balls with the same 6 IDs, etc.).
    We both do the fun activity to see whether there's any difference in our results.


    Question: Are there different (distinguishable) possible results from doing a fun activity repeatedly, i.e., more than one "way to label the balls"?
    Answer: Yes. Suppose we both have a ball with ID 123-45-6789. Perhaps by chance you labelled yours IN-THE-BOX, but I labelled mine OUTSIDE-THE-BOX. This certifies that your result differs from mine.

    Question: So how many (distinguishable) results are possible when doing a fun activity?
    Answer: 6C4.

    Question: If somebody does the fun activity as above, but instead writes "IN-THE-BOX" on 2 labels and "OUTSIDE-THE-BOX" on 4 lables, then how many results are there?
    Answer: 6C2.


    Now think about why these last two answers must be the same. If it helps, imagine writing LONDON on 4 labels and PARIS on 2 labels...
    What's important is that some 4 labels are indistinguishable from one another, the other 2 labels are likewise indistinguishable from one another, but any one of the first 4 labels is distinguishable from any one of the other 2 labels.

    ReplyDelete
  2. GlenApril 23, 2011 at 6:11 AM

    I'll try a shorter explanation.

    Whether you choose 2 of 6 or 4 of 6, you do it by separating the 6 into two piles, one of 2 and one of 4. There are a certain number of ways you can do that. Whether you consider the 2-pile or the 4-pile the "chosen ones" doesn't change that number.

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  3. Catherine JohnsonApril 24, 2011 at 6:36 PM

    Glen - thank you!

    Anonymous - I'm going to have to come back and read this tomorrow morning when I'm fresh!

    ReplyDelete
  4. AnonymousApril 26, 2011 at 8:23 AM

    http://www.flickr.com/photos/36762583@N04/5658060190/

    ReplyDelete
  5. AnonymousApril 26, 2011 at 12:16 PM

    an improved version of the last link.

    ReplyDelete