Question

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

Answer 1

Hint: In order to solve this question, we will form the groups on the basis of common difference and to do so, we will try to form AP for different common difference for \[\dfrac{12}{3}=4\] elements like d = 0, 1, 2, … and will continue forming them until we get the fourth element greater than 12 for the AP of the first term 1.

Complete step-by-step answer:
In this question, we have to find the number of ways in which we can divide the first 12 numbers into three different groups such that the number in each group will be in AP. To solve this, we will form the groups on the basis of the common difference and to do so, we will try to form AP for different common difference for \[\dfrac{12}{3}=4\] elements like d = 0, 1, 2,…. Here we have taken 4 elements in each group because we have to divide 12 natural numbers into 3 groups.
Now, for the common difference, d = 0, we can form AP like (1, 1, 1, 1), (2, 2, 2, 2), (3, 3, 3, 3)….. (12, 12, 12, 12), that are 12 ways.
Now, for common difference, d = 1, we can form AP of 4 elements like (1, 2, 3, 4), (2, 3, 4, 5), (3, 4, 5, 6)…..(9, 10, 11, 12), that are 9 ways.
Now for common difference d = 2, we can form AP of 4 elements like (1, 3, 5, 7), (2, 4, 6, 8), (3, 5, 7, 9), …. (6, 8, 10, 12), that are 6 ways.
Now for common difference d = 3, we can form AP of 4 elements like (1, 4, 7, 10), (2, 5, 8, 11), (3, 6, 9, 12)., that are 3 ways.
Now for common difference, d = 4, we can form AP of 4 elements like (1, 5, 9, 13). Here, we get the last term greater than 12. So, AP of such a case is not possible.
Therefore, we can say there are 4 possible groups of forming AP, that are d = 0, 1, 2, 3. Hence, there are 4 different ways in which the first 12 natural numbers can be divided into a group of 3, and elements of each group will be in AP.

Note: The possible mistake one can make while solving this is counting the number of APs that can be formed which is wrong. Here, we have to count the number of ways in which AP can be formed that is, the groups on the basis of common difference.