Why do you multiply 4 choose 2 and 4 choose 2 together and not just add them?

  • ADMIN M0★ M1 M5

    [Originally posted by a student in the Discussions, before it was replaced by this forum.]

    Module 0 Week 2 Day 8 Your Turn Explanation

  • ADMIN M0★ M1 M5

    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).

    M0W2D8-y-choose-horizontal-vertical-lines.png

    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.)

    M0W2D8-y-choose-horizontal-vertical-lines2.png

     

    (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.