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January 5th, 2013, 04:17 AM   #1
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A fairly basic counting problem that I'm drawing a blank on.

Hi all, I hope I'm posting this in the correct forum:

The question is this: An ice cream store has 20 different flavours. In how many ways can we order a dozen different ice cream cones, if each cone has 2 different flavours ?

If it was just 12 different cones from 20 different flavours, I would go C(20,12) = 125,970 but it's the "if each cone has 2 different flavours that I'm drawing a blank on. I imagine I'll feel like an idiot when it's pointed out, but any help would be much appreciated. Thank you.
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January 5th, 2013, 08:56 AM   #2
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Re: A fairly basic counting problem that I'm drawing a blank

i believe you just have to multiply that number with 20*19 because that's the amount of ways you can get 2 different flavours
and divide by 2 since all combinations have been taken twice
(such as {lemon,vanilla} is the same as {vanilla, lemon} )
so 125,970*190= 23,934,300
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January 5th, 2013, 09:00 AM   #3
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Re: A fairly basic counting problem that I'm drawing a blank

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January 5th, 2013, 10:19 AM   #4
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Re: A fairly basic counting problem that I'm drawing a blank

This is what I think. First choose 12 from 20, which is 20_C_12. Next, for each cone you have 19 degrees of freedom left so multiply by 19^12. Answer should be (20_C_12) x (19^12).
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January 5th, 2013, 10:55 AM   #5
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Re: A fairly basic counting problem that I'm drawing a blank

Sorry, I was in error. Mr Zardoz has it correct
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January 5th, 2013, 02:19 PM   #6
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Re: A fairly basic counting problem that I'm drawing a blank

summsies!

The word "different" makes the problem trickier.


Quote:
An ice cream store has 20 different flavours.
In how many ways can we order a dozen [color=blue]different[/color] ice cream cones, if each cone has 2 different flavours?







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January 5th, 2013, 09:23 PM   #7
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Re: A fairly basic counting problem that I'm drawing a blank

Quote:
Originally Posted by soroban
summsies!




i calculated that if u need it in numbers.
it should be 3345763705175979360
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January 6th, 2013, 01:12 AM   #8
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Re: A fairly basic counting problem that I'm drawing a blank

Thanks to all who replied: Soroban, yes it was definitely the wording of different cones each with different flavours that tripped me up. These sneaky Math profs, hey So just to recap so I understand: From 20 different flavours, we're choosing 2 flavous per cone, giving us 190 possible combo's there, and then from that 190 possible combinations, we're making 12 different cones. I think that makes sense. Thanks again to all for your input !
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January 6th, 2013, 01:21 AM   #9
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Re: A fairly basic counting problem that I'm drawing a blank

Quote:
Originally Posted by ZardoZ
Zardoz - In your answer, is your line of reasoning: 20 ways to choose the first flavour, 19 ways to choose the second, divide by 2 different flavours = 190 ^12 because there are 12 different cones being made? Both yours and Soroban's solutions make sense to me!
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January 6th, 2013, 02:11 AM   #10
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Re: A fairly basic counting problem that I'm drawing a blank

[color=#000000]Soroban's solution is correct![/color]
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