Find the probability that the string includes 2 consecutive elements. Four distinct digits are selected at random from the set 0, 1, 2, . . . , 9.
Find the probability that the selection includes two consecutive digits.

First, I found the total number of ways to select $4$ digits from the set, that is, ${\binom{10}{4}}$. Then I found the number of ways to select 4 digits from the set so that there are no consecutive integers, that is, ${\binom{7}{4}}$, and subtracted it from ${\binom{10}{4}}$.
So my answer is ${\binom{10}{4}} - {\binom{7}{4}}$ divided by ${\binom{10}{4}}$. Is this correct?
Edit: I thought the number of ways to select 4 digits from the set so that there is no consecutive element is the same as the number of ways to rearrange $AAAABBBBBB$ so that there are no consecutive $A$s. If I take the string $BBBBBB$ then I notice that there are 7 places to insert the 4 $A$s I have. The number of rearrangements is ${\binom{7}{4}}$. So the number of ways to select 4 digits so that there are no consecutive digits is ${\binom{7}{4}}$. 
 A: Your $AAAABBBBBB$ permutation is effectively a selection mask from $0123456789$.
To get a selection mask with no consecutive $A$'s, you can set up a skeleton like this: $\_\_AB\_\_AB\_\_AB\_\_A\_\_$ then add the remaining 3 $B$'s into the 5 permitted zones, which is a "stars&bars" category assignment of $5+3-1=7$ choose $3, {7\choose 3}={7\choose 4}$ consistent with your result.
Boiling down the probability into factorials, 
$$ \frac{{7\choose 3}}{{10\choose 4}}  = \frac{7!}{3!4!}\frac{6!4!}{10!} = \frac{6\times 5\times 4}{10\times 9\times 8} = \frac{1}{6}$$
A: The main difficulty in this problem is to compute the number of ways to select $4$ nonconsecutive digits. The explanation why $7 \choose 4$ is the correct answer: one may notice that it's enough to choose $4$ from $7$ labeled items and then add one more after each chosen item (except for the last). This will make us sure that we have selected non consecutive digits (because there is a correspondence between the items and digits) and there are ten items/digits in total.
The description might be not clear, so here's an example. Assume that the items are letters A, B, C, D, E, F, G. We choose A, B, D and G, add an additional letter after A, B, D. Now we have: A, _, B, _, C, D, _, E, F, G. Ten symbols, four selected, none of them consecutive.
