In how many different ways can the first 12 natural numbers be divided into three different groups such that numbers in each group are in A.P.?

No group of four numbers from the first 12 natural numbers can have the common difference 4.If one group including 1 is selected with the common difference 1, then the other two group can have the common difference 1 or 2If one group including 1 is selected with the common difference 2, then one of the other two groups can have the common difference 2 and the remaining group will have common difference 1. If one group including 1 is selected with the common difference 3, then the other two groups can have the common difference 3.Therefore, the required number of ways is 2 + 1 + 1 = 4.