Combinatorics - Probability of given two cards next to each other (Total cards = 13) 
There is a set of 13 cards, numbered from 1 to 13. What is the probability that when these cards are randomly laid out in a line, Card-1 and Card-2 will be next to each other?
The given answer (from CK12.org) is   $1/{13 \choose 2}=\frac{1}{78}$ which does not make sense to me.
My answer is;

*

*Total permutations = $13!$

*Permutations with Card-1 and Card-2 next to each other = Put card-1 and card-2 to a bigger single card, so there are 12 cards, therefore $12!$ permutations. However, within the bigger card, there are $2!$ permutations, so  $2\cdot12!$ permutations where card-1 and card-2 are next to each other.

Therefore the probability = $\frac{2\cdot12!}{13!}=\frac{2}{13}$.
Can anyone help me understand where I have gone wrong?
 A: You have not gone wrong; the answer sheet has.
The probability for obtaining the adjacency (as you argued) when selecting any two among thirteen places is: $\left.{^{12}{\mathrm C}_1}\middle/{^{13}{\mathrm C}_2}\right.$ which is indeed $\tfrac{2}{13}$ .
A: These are other ways to compute the probability:
$12$ cards (excluding Card - 2) are laid out randomly in a line. You then put Card - 2 between any two cards or at either ends of the line so there are $13$ possibilities. Out of these, only if you put Card - 2 on the left or right of Card - 1 will they be next to each other. Probability $\frac{2}{13}$
Let $a$ denote the number of cards to the left of both cards in interest, $b$ the number of cards between them, and $c$ the number of cards to the right of both of them. $a+b+c=11,\phantom{x}a,b,c\geq 0$. Using stars and bars we get $\binom{11+3-1}{3-1}$ possibilities. If the cards are next to each other, there won’t be any cards between them i.e. $b=0\leftrightarrow a+c=11,\phantom{x}a,c\geq0$. Stars and bars give $\binom{11+2-1}{2-1}$ possibilities. Probability $\binom{11+2-1}{2-1}/\binom{11+3-1}{3-1}=\frac{2}{13}$.
I prefer the later method because can use it for more complicated problems e.g. if we want to calculate probability of any 2 of 5 cards to be next to each other if there are a total of 20 cards.
