What do you call this symbol?
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[Originally posted in Discussions]
It’s the 6 over the 2 in parentheses
That's a good question! This symbol,
means "the number of ways to choose two things out of six things."
It's easy to mistakenly think this should be 6 × 5, because we learned before that the ways to line up six things in a row is equal to 6! = 6 × 5 × 4 × 3 × 2 × 1, and this looks similar, except you are not taking all of the six things, only two of them.
But.... it's not equal to 6 × 5! This is because "choose two things out of six things" doesn't imagine you putting them in a line. It just imagines them all mixed up. The 6 × 5 ways double-counts every way. For example, if you were choosing A and B out of six things, the ways AB and BA are both counted in 6 × 5.
This is why the number of ways to choose two things out of six things is really 6 × 5 × ½.
If you can be very careful and keep track of when order matters (i.e. AB and BA are different ways), and when order doesn't matter (i.e. when AB and BA count as the same way), then you're well on your way to understanding a lot of great combinatorial problems!
The Daily Challenge Team