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

]]>Prof Loh does it in Module 3, Day 4 Challenge Question ]]>

Thatβs getting into combinatorics in a geometry question explanation lol ]]>

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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