Solution
Hidden Progressions
Show the problem again
How many 3-element subsets of {1, 2, ..., 20} have the property that, when their elements are listed in increasing order a < b < c, they form an arithmetic progression (that is, b − a = c − b)? Give an integer.
Worked solution
The answer is 90. The condition is a + c = 2b. Given the two outer terms a and c, the middle term b = (a + c)/2 is forced, and it is an integer strictly between them exactly when a and c have the same parity. So such subsets correspond one-to-one with pairs {a, c} of distinct numbers of equal parity. There are 10 odd and 10 even numbers in range, giving 2 × C(10, 2) = 2 × 45 = 90.
Source: Classic counting exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.