Why do you multiply 4 choose 2 and 4 choose 2 together and not just add them?
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[Originally posted by a student in the Discussions, before it was replaced by this forum.]
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Good question! The reason you multiply \(\binom{4}{2}\) and \(\binom{4}{2}\) can be visualized by imagining groups: first, choose the two vertical lines (let's call them A, B, C and D).
There are 6 ways to do this:
$$ AB\\\\ AC\\\\ AD\\\\ BC\\\\ BD\\\\ CD\\\\ $$For each of these 6 ways there are 6 other ways to choose the horizontal lines (let's call them 1, 2, 3 and 4.)
(6 ways starting with AB, 6 ways starting with AD, 6 ways starting with BC, etc.)
AB 12 AC12 AD12 BC12 BD12 CD12
AB 13 AC13 AD13 BC13 BD13 CD13
AB 14 AC14 AD14 BC14 BD14 CD14
AB 23 AC23 AD23 BC23 BD23 CD23
AB 24 AC24 AD24 BC24 BD24 CD24
AB 34 AC34 AD34 BC34 BD34 CD34
There are six groups, with six ways in each group, so that is why there are \(6 \times 6 =36\) ways rather than just \( 6 + 6 = 12\) ways.
