Combination/Permutation Problem: a piece of paper has 10 distinct dots drawn on it. If a pair of do

[Hammer Time]

The question is this: a piece of paper has 10 distinct dots drawn on it. If a pair of dots determines a line, how many different ways are there to draw a line on this paper?

This question is from math destroyer. I'm having a hard time knowing whether order counts or whether it doesn't, and why that matters and how that changes the answer. I've watched the Chad's video about it. But I don't think he did a very good job explaining it because I always seem to get these questions wrong.

So I've looked up the answer, and it's apparently 10!/8!*2!
What???? Where does the 8! come from?? Where does the 2 come from????? Why are we dividing by anything at all????
I can understand that making a line from A-B is the same as B-A. So I guess i can understand dividing by two. But what's the deal with the 8!?????? What do they mean by order matters here??

Thanks in advance 😛

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[cdp1210]

This is a combination problem; order does not matter. These are important because they can be worded different but its always the same concept. Imagine the question being worded "How many ways can you select 2 dots out of 10" as 2 dots signify a line. Same thing as how many way can you select 2 people out of a group of 10 people. Why does order not matter for this? Take for example you have a group of 10 people. Let say you select 2 people (Paul and John). Does it matter if you select Paul THEN John vs John THEN Paul? No. It's the same 2 people being selected either way so order does NOT matter (hence a combination problem).

Bring that back to the dots and line problem. Image each dot was numbered 1-10. If you were drawing a line between dot #1 and #5 for example would it matter if you started that line at #1 and drew it through #5 versus drawing it in the other direction? No. A line going #1-->#5 is the same as a line going from #5-->#1 therefore order does not matter and it is a combination problem.

As far as 10!/8!*2! that is the formula for combination problems. n!/k!(n-k)!
n=10 (how many you are choosing from)
k=2 (how many you are choosing)
n-k=8 (the difference)

I personally like to simplify it down to (10 choose 2) which would be (10x9/2x1)=45. There is lots of information online to learn more about combinations and permutations if you need more clarification, but it is an important topic to learn.

 

[thegreat267]

This is a combination problem; order does not matter. These are important because they can be worded different but its always the same concept. Imagine the question being worded "How many ways can you select 2 dots out of 10" as 2 dots signify a line. Same thing as how many way can you select 2 people out of a group of 10 people. Why does order not matter for this? Take for example you have a group of 10 people. Let say you select 2 people (Paul and John). Does it matter if you select Paul THEN John vs John THEN Paul? No. It's the same 2 people being selected either way so order does NOT matter (hence a combination problem).

Bring that back to the dots and line problem. Image each dot was numbered 1-10. If you were drawing a line between dot #1 and #5 for example would it matter if you started that line at #1 and drew it through #5 versus drawing it in the other direction? No. A line going #1-->#5 is the same as a line going from #5-->#1 therefore order does not matter and it is a combination problem.

As far as 10!/8!*2! that is the formula for combination problems. n!/k!(n-k)!
n=10 (how many you are choosing from)
k=2 (how many you are choosing)
n-k=8 (the difference)

I personally like to simplify it down to (10 choose 2) which would be (10x9/2x1)=45. There is lots of information online to learn more about combinations and permutations if you need more clarification, but it is an important topic to learn.

Great explanation.

 

[Hammer Time]

This is a combination problem; order does not matter. These are important because they can be worded different but its always the same concept. Imagine the question being worded "How many ways can you select 2 dots out of 10" as 2 dots signify a line. Same thing as how many way can you select 2 people out of a group of 10 people. Why does order not matter for this? Take for example you have a group of 10 people. Let say you select 2 people (Paul and John). Does it matter if you select Paul THEN John vs John THEN Paul? No. It's the same 2 people being selected either way so order does NOT matter (hence a combination problem).

Bring that back to the dots and line problem. Image each dot was numbered 1-10. If you were drawing a line between dot #1 and #5 for example would it matter if you started that line at #1 and drew it through #5 versus drawing it in the other direction? No. A line going #1-->#5 is the same as a line going from #5-->#1 therefore order does not matter and it is a combination problem.

As far as 10!/8!*2! that is the formula for combination problems. n!/k!(n-k)!
n=10 (how many you are choosing from)
k=2 (how many you are choosing)
n-k=8 (the difference)

I personally like to simplify it down to (10 choose 2) which would be (10x9/2x1)=45. There is lots of information online to learn more about combinations and permutations if you need more clarification, but it is an important topic to learn.

Thanks, I like thinking of it as there are ten dots to choose from and you are taking two. And since order doesn't matter, I can see why you divide by 2!. I was thinking of it in the wrong way, thanks again 😀

 

[alexis2010]

You can think of it as a combination problem or you can do it this way: from the first for you can draw 9 lines. From the second dot you can only draw 8 lines, from the third dot you can draw 7 lines etc...
9+8+7+6+5+4+3+2+1= 45