What is the number of ways to select ten distinct letters from the alphabet $\{a, b, c, \ldots, z\}$, if no two consecutive letters can be selected? 
What is the number of ways to select ten distinct letters from the alphabet $\{a, b, c, \ldots, z\}$, if no two consecutive letters can be selected?

There are a couple of formulas that I could use, but I am not sure how to use them.

The number of combinations of a given length that use elements from a given set, allowing repetition and with no missing elements in other words, each element of the given set occurs at least once in each combination: $\binom{k-1}{n-1}$ where where $n$ is the number of elements in the given set and $k$ is the length of the combination.

OR

The number of combinations of a given length that use elements from a given set, allowing repetition: $\binom{n+k-1}{n-1}=\binom{n+k-1}{k}$ where $n$ is number of elements in the given set and $k$ is the length of the combination.

Please help me understand what I should do and how to go about solving this question?

A similar question was asked a while ago, where $r$ non-consecutive numbers had to be selected from $n$ numbers. Basically, the problem at hand can be converted to a stars-and-bars problem with $11$ slots and $10$ bars, where the bars represent the indices of the selected letters. The remaining 16 letters must be divided among the $11$ slots, with at least one letter in each of the $9$ slots in the middle. Once one letter is placed in each of these slots, there are $26 - 10 - 9 = 7$ letters left. As such, the number of ways to select $10$ non-consecutive letters from the alphabet equals:
$${7 + 10 \choose 10} = {17 \choose 10} = 19448$$
