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?

Here is another ways of visualizing the problem.  First, we arrange $16$ blue balls and $10$ green balls so that no two of the green balls are consecutive. Then we label the balls. 
Line up $16$ blue balls, leaving spaces between successive balls and at the ends of the row.  There are $17$ such spaces, $15$ between successive blue balls and two at the ends of the row.
$$\square \color{blue}{\bullet} \square \color{blue}{\bullet} \square \color{blue}{\bullet} \square \color{blue}{\bullet} \square \color{blue}{\bullet} \square \color{blue}{\bullet} \square \color{blue}{\bullet} \square \color{blue}{\bullet} \square \color{blue}{\bullet} \square \color{blue}{\bullet} \square \color{blue}{\bullet} \square \color{blue}{\bullet} \square \color{blue}{\bullet} \square \color{blue}{\bullet} \square \color{blue}{\bullet} \square \color{blue}{\bullet} \square$$
To ensure that no two of the ten green balls are adjacent, we choose $10$ of these $17$ spaces in which to insert a green ball, which can be done in 
$$\binom{17}{10}$$ 
ways.  Next, we label the balls from left to right.  The letters on the green balls are the desired subset of the alphabet in which no two of the selected letters are consecutive.  For instance, 
$$\color{green}{\bullet} \color{blue}{\bullet} \color{green}{\bullet} \color{blue}{\bullet} \color{blue}{\bullet} \color{blue}{\bullet} \color{green}{\bullet} \color{blue}{\bullet} \color{blue}{\bullet} \color{green}{\bullet} \color{blue}{\bullet} \color{green}{\bullet} \color{blue}{\bullet} \color{green}{\bullet} \color{blue}{\bullet}  \color{blue}{\bullet} \color{blue}{\bullet} \color{green}{\bullet} \color{blue}{\bullet} \color{green}{\bullet} \color{blue}{\bullet} \color{green}{\bullet} \color{blue}{\bullet} \color{green}{\bullet} \color{blue}{\bullet}  \color{blue}{\bullet}$$
corresponds to the selection $\{A, C, G, J, L, N, R, T, V, X\}$.
