Combinations

To solve the problem, we start by considering the condition given: , , and are in arithmetic progression (AP). In an AP, the difference between consecutive terms is constant. Using the combination formula , we have:



Now calculate each combination:




Substitute into the AP condition:


Simplify the expressions:
Factor out from the left side:


Now the equation becomes:


After evaluating the possibilities for , it turns out that when , the condition holds, and both sides of the equation will be balanced.
Therefore, the value of is 7.