Find the probability that four natural numbers selected from 1 to 20 forms an A.P.

Ans is 57/(20c4)= 1/85

First totally the max common difference possible for the 4 nos in ap is 6 (1,7,13,19) and(2,8,14,20) which comes out to be 2 ways..then similarly for common difference 5 it comes out to be 5 possibilities now u must notice that the no of possibilities itself are in ap with c.d = 3,so total no of possibilities is nothing but sum of no of now in 1-20 having cd 6,5,4...1.that is 2+5 +8+...17  = 57  which is numerator and denominator is no of ways of selecting 4 nos from 20 that is 20c4..hope this helps