Reading and Understanding profile bands in DNA Fingerprinting
How you would determine the plasmid map from just the restriction digest data

The thought process behind doing this is a bit involved.


From the question you are told that the Gene is 8kb and does not contain an EcoRI site inside of the gene


I say not inside because the ends of the gene were made with EcoRI cuts, but digesting in EcoRI will only produce the 8kb fragment of the gene

The vector is 6kb and you know that they have EcoRI sites at either end, but you are not told what other cut sites there are.
When you digest the plasmid in BamHI you get 3 fragments, meaning that your plasmid has 3 BamHI sites somewhere in it.
You know that your total plasmid size is 14kb (easiest way to know is just look and see that they tell you that there is an 8kb gene and a 6kb vector). Looking at the Double Restriction Digest, you can see that the five fragments add up to 14kb, which implies that there are only one of each fragment in the plasmid, and they are produced by 5 separate cuts.


This is an important piece of information because this tells you that you do not have 2 fragments of the same size making up different parts of the plasmid.
Remember that when you do a restriction digest on a circular piece of DNA, the number of fragment you get equals the number of cuts (restriction sites) in the plasmid. So because there were 5 fragments, you have 5 separate restriction sites
From the Double Digest data, you also know that the smallest fragment of DNA that you have is 1kb and the largest is 4kb.

Since the 4kb site is common to both the BamHI only and the the Double Digest, as since you do not have a 0.5kb or 1.5kb fragment (the remainders if the Double Digest 4kb fragment had been produced from the 4.5kb or 5.5kb fragments in the BamHI digest) in the Double Digest, you know that the 4kb fragment is the same in both the BamHI digest and the Double Digest, and is produced by 2 BamHI cuts.


You can now work on roughly placing where the 4kb fragment should be in the map.
If you try to place that 4kb fragment in the vector, then you would either generate a 2kb fragment, or something less than 2kb and a very small fragment.
Since you do not have a 2kb fragment or fragments smaller than 1kb, you know that it cannot be in the vector region. Also in order to get a 2kb fragment from the 6kb vector the 4kb fragment would have needed to be cut once with EcoRI and from the description, above, we worked out that the 4kb fragment is produced by two BamHI cuts
It could have been that the vector was 1kb + 4kb + 1kb, however from point 4 you determined that you only had one of each fragment, so you can't have two 1kb fragments.

All of that tells you that it has to be in the 8kb gene portion, though you need some more information to determine exactly where. 


Remember that you do know that the 4kb fragment was produced by 2 BamHI cuts, which means that it cannot be at the very end of the gene region, as that would require it to be cut once with EcoRI
This also tells you that you have at least two BamHI cut sights in the gene.

What I would do here is draw two circles, the first having an 8kb arc and a 6kb arc, then draw second circle and make three cuts in it. 


From 6 we worked out that the 4kb piece will overlap somewhere in the 8kb area, that it cannot be at an end and that there need to be 2 BamHI restriction sites within the gene. 
You also have to consider the 4.5kb and 5.5kb parts from the BamHI digest. Since 4kb plus either of those two are greater than 8kb, then you know that the cut that separates 4.5kb from 5.5kb will be somewhere in the 6kb vector. This is how you know that the third BamHI site is in the vector.

Now that you have determine that the 4kb fragment is in the gene you can start to work out the other pieces.
You had 2 fragments from the BamHI digest that were 5.5kb and 4.5kb and in the double digest you have fragments that are 1.0kb, 2.5kb, 3.0kb, and 3.5kb.


Since you have already determine that that 4kb piece is the same for both digests, then at this point you can tell yourself that the 1.0kb and 3.5kb fragments made up the 4.5kb fragments and the 2.5kb and 3.0kb fragments made up the 5.5kb fragment.

Since you know that there has to be two EcoRI cuts that make the 6kb fragment, you are going to need two fragments that add up to 6kb.


The only way to get this with the four fragments that you have from the double digest is with the 2.5kb piece and the 3.5 kb piece.
So you have now determined the exact position of one of the BamHI cut sites. It is in the Vector, 2.5kb from one of the EcoRI sites and 3.5kb from the other EcoRI sites.

Now that you determined where one of the sites is, you know that the 2.5kb fragment was paired with the 3.0kb fragment, so this tells you that there is a BamHI site 3kb away from the EcoRI site in the Gene.
From there you can connect the 4.0kb piece to the end of the 3.0kb piece because you know that the 3.5kb piece has to be attached to the 1kb piece and cannot be attached to the 3kb piece.
All that is left is to close the circle. You know that you have 1.0kb more of the gene that needs to be place and this will connect the 7.0kb that you determine of the gene in step 11 to the 3.5kb fragment in the vector. You have that 1.0kb fragment, and that is the one.


So that is how you determine how to construct the plasmid map from the restriction digests.  

What the distances between the cut sites tell you

The cut sites are determined by the sequence of the DNA that the Restriction Enzyme recognizes. 


EcoRI recognizes GAATTC
BamHI recognizes GGATCC


You will notice that these are palindromic sequences, so 5'-GAATTC-3' is 3'-CTTAAG-5' on the complimentary strand. 

All that the distances are telling you is that in the areas in between the cut sites, the sequences GAATTC and GGATCC do not appear. If they did, then the Restriction Endonuclease would recognize that position and cut the DNA, and you would have an additional cut site at that location.

Assuming that you were given the plasmid map, this should be the answer. 

The answer should be C. The 1kb and 3kb fragments are the DNA fragments from the gene that are generated from the EcoRI / BamHI digest. So one end of the fragment was produced with EcoRI and the other end of the fragment was produced with BamHI 

The 4kb fragment is the result of BamHI restriction alone. So both ends of the fragment were produced by BamHI so it cannot be the answer to a fragment produced by EcoRI / BamHI restriction. 

If they are saying that they also want answer A as well, then their question is very poorly worded, in my opinion.