Which answer is correct?

Steven G

Steven G

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How many different ways are there to tile a 2 by 8 rectangle with identical 2 by 1 dominoes so that exactly 2 of the dominoes are vertical?

I can see why the answer is 5C2 =10 (3 sets of horizontal and two vertical) but I do not see why it is not 8C2 (there are 6 horizontal dominoes and 2 vertical or there are 3 sets of horizontal and 8 possible places to pick for the two vertical dominoes). Can someone please enlighten me?
 
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Why is it 5C2? Why are there 8 horizontal dominoes? I see only 6.

Let's see. There are 4 places for the vertical dominoes: 1, 2, 3, 4. If we count 2 digit combinations we'll get too many since 12 and 21 are the same (dominoes are identical). So, let's pick 1st domino's position: 1 through 4. The second can be in the same position or to the right of the first: 11, 12, 13, 14; 22, 23, 24, etc. That's n(n+1)/2 = 10.
 
Jomo said:
e6e4d7ea8bd719e2a87c10969def758098910e2c.png
How many different ways are there to tile a 2 by 8 rectangle with identical 2 by 1 dominoes so that exactly 2 of the dominoes are vertical?

I can see why the answer is 5C2 =10 (3 sets of horizontal and two vertical) but I do not see why it is not 8C2 (there are 8 horizontal dominoes and 2 vertical or there are 3 sets of horizontal and 8 possible places to pick for the two vertical dominoes). Can someone please enlighten me?
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There are only 8 dominoes in total. If exactly two are vertical, then the remaining 6 are in 3 horizontal pairs. 2 + 3 = 5 positions.
 
lev888 said:
Why is it 5C2? Why are there 8 horizontal dominoes? I see only 6.

Let's see. There are 4 places for the vertical dominoes: 1, 2, 3, 4. If we count 2 digit combinations we'll get too many since 12 and 21 are the same (dominoes are identical). So, let's pick 1st domino's position: 1 through 4. The second can be in the same position or to the right of the first: 11, 12, 13, 14; 22, 23, 24, etc. That's n(n+1)/2 = 10.
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Why is it 5C2? There are 5 columns of dominoes (3 horizontal (totaling 6 dominoes) and 2 vertical--so 5C2)
Why are there 8 horizontal dominoes? I don't think that I said that. There are 8 dominoes and I want to make 2 of them different from the other 6, hence 8C2.

Ok, so yes you showed me how to see its 10 and convinced me that it is 10 and I thank you for that. My concern is that if someone else claimed it was 8C2 I can not explain to them why it is wrong other than showing them that it must be 5C2. I am sure Dr P would say to try to list the 8C2 ways and see where it goes wrongs (thank you!). I will work on that later today.
 
Jomo said:
Why is it 5C2? There are 5 columns of dominoes (3 horizontal (totaling 6 dominoes) and 2 vertical--so 5C2)
Why are there 8 horizontal dominoes? I don't think that I said that. There are 8 dominoes and I want to make 2 of them different from the other 6, hence 8C2.

Ok, so yes you showed me how to see its 10 and convinced me that it is 10 and I thank you for that. My concern is that if someone else claimed it was 8C2 I can not explain to them why it is wrong other than showing them that it must be 5C2. I am sure Dr P would say to try to list the 8C2 ways and see where it goes wrongs (thank you!). I will work on that later today.
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Sorry, I read "there are 8 horizontal dominoes" and assumed you referred specifically to horizontal dominoes.
Ok, I understand the 5C2 approach.
Regarding 8C2 - it is the right answer to the question "how many ways are there to pick 2 dominoes out of 8?". But it's not the question you posed. Shouldn't the answer address the tiling part of the question?
 
JeffM said:
There are only 8 dominoes in total. If exactly two are vertical, then the remaining 6 are in 3 horizontal pairs. 2 + 3 = 5 positions.
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JeffM, That was exactly where I was not seeing things. But now I see it crystal clear. The horizontal are in pairs, ie you can't put a vertical in between a pair. It as if each pair is taped together and is counted as one. Not sure why this bothered me. Even though I wrote pairs in my posts, reading it in your post made the light bulb come on! Thanks
 
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