Should same-angles but opposite order combinations be allowed?

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    Module 2 Week 2 Day 8 Challenge Part 3

    Whether repeat (like same angles but opposite order) combinations are allowed should be specified in mini-question

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    @The-Rogue-Blade I don't completely get what you're asking. (assuming you are talking about the opposite order of lines)I think that repeats are not allowed because that would be the same thing. I considered each angle being surrounded by a pair of lines and never thought of the order.
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  • ADMIN M0★ M1 M5

    @The-Rogue-Blade I see what you're asking! You're saying that perhaps this pair of angles should be counted twice, depending on which is the first angle and which is the second angle:

    M2W2D8-ch-part-3-question-pairs-lines-need-specify-order.png

    The double-counting occurs if order matters. For example, in questions like, "How many ways are there to choose two out of five people to stand in a line?" 👩‍👩‍👧 👨‍👨‍👧 or "How many ways are there to make an ice cream cone with two scoops if you can choose from vanilla, chocolate, mocha, mint chip, or raspberry?" 🍦 🍦 etc., then order matters.

    If we say "How many ways are there to choose a pair of students out of 5 to sing in the performance?" 🎵 then, does order matter? No, it doesn't.

    We have a convention of counting "the ways to choose two things out of \(n\) things," and it's called a "binomial coefficient," which looks like \( \binom{n}{2}\), or it's called more casually a "choose." You will see more of this in Module 3: Combinatorics. 🙂 We also saw chooses in Module 0 with the Day 8 lesson about counting the number of triangles out of a bunch of criss-crossing lines. In that lesson, we saw

    $$ \text{ number of ways to choose } 2 \text{ lines out of } 6 \text{ lines } = \frac{6 \times 5 }{2} $$

    We divided by \(2\) because order didn't matter; whether we chose a line first or second didn't affect the answer.

    This mini-question is asking for the number of pairs in the same way, so that's why we don't count the highlighted \(40^{\circ}\) and \(20^{\circ}\) angles twice as in the diagram above.

    🙂

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    @debbie
    That’s getting into combinatorics in a geometry question explanation lol 🙂

  • ADMIN M0★ M1 M5

    @RZ923 I wonder if it's as easy to put geometry into a combinatorics problem....... 🙂

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    @debbie
    Prof Loh does it in Module 3, Day 4 Challenge Question 🙂