I have a set of 11 numbers {0,3,6,9,12,15,18,21,24,27,30}. I am currently grouping numbers with a spacing of 9. I did this by hand - {0,9},{3,12},....{21,30} of total 8 pairs.
The answer is 8. But I am not sure how to relate this mathematically - or maybe there are another set pairs, which I might have missed. I am definitely sure this is related to permutations and combinations, but I don't know how to relate it exactly. Any guidance will be helpful.
Thank you.
Your method is essentially correct. You are sure to find all the pairs that differ by $9$ if you look ahead $9$ from each number in the list to see if the result is in the list. You don't need anything fancy about permutations and combinations.
In this particular example the list is an arithmetic progression and $9$ is three times the spacing, so you know you will find a difference of $9$ for all but the last $3$ numbers on the list. That's why there are $11 - 3 = 8$ pairs.