How many ways to select $r$ items from $n$ items, without selecting any two consecutive items? Suppose that there are $n$ terms which are numbered from $1$ to $n$. We have to select any $r$ terms out of $n$. But there shouldn't be any two consecutive terms. In how many ways this can be done?
E.g: There are $30$ balls which are numbered from $1$ to $30$. We are commanded to select $5$ balls, but any two of them should not have consecutive numbers on them. In how many different ways can we do this?
 A: For the specific example of selecting $5$ numbers from the set $\{1, 2, 3, \ldots, 30\}$ so that no two consecutive numbers are selected, line up $25$ blue balls, leaving gaps between them and at the ends of the row.  There are $25 - 1 = 24$ spaces between successive blue balls and two spaces at the ends of the row for a total of $26$ spaces in which to place the green balls.  We can fill five of these spaces with green balls in $\binom{26}{5}$ ways.  Now number the balls from left to right.  The numbers on the green balls are the desired set of five numbers in which no two of the numbers are consecutive.
For the general case, line up $n - r$ blue balls.  That leaves $n - r + 1$ spaces ($n - r - 1$ spaces between successive blue balls and the two spaces at the ends of the row) in which to place the green balls.  Provided that $r \leq n - r + 1$, we can fill $r$ of these $n - r + 1$ spaces with green balls in $\binom{n - r + 1}{r}$ ways.  Hence, the number of subsets of $r$ elements selected from a set with $n$ elements so that no two consecutive elements are chosen is 
$$\binom{n - r + 1}{r}$$
