Question

Using an ordered alphabet of 26 letters, how many ways are there to choose a set of six letters such that no two letters in the set are adjacent in the alphabet?
For instance, {ISOKAY} is a valid set of six numbers, but {VETOIF} is not because E and F are in the same set.

Answer 1

To find our solution, we can start off by creating a string of 27 boxes, all followed by the letters of the alphabet. Underneath the boxes, we can place 6 pairs of boxes and 15 empty boxes.The stars represent the six letters we pick. The empty boxes to the left of the stars provide the "padding" needed to ensure that no two adjacent letters are chosen. We can create this - 
$\binom {21} {6}= \frac{21*20*19*18*17*16}{6*5*4*3*2*1} =21*19*17*8=54264$
Thus, the answer is that there are $\boxed{54264}$ ways to choose a set of six letters such that no two letters in the set are adjacent in the alphabet. Hope this helped and have a phenomenal New Year! 2018