Eeny, meeny, miny, moe

The integers \(1,2,\ldots, 17\) are divided into 5 disjoint sets. One set has 5 elements, one set has 4 elements, two sets have 3 elements and the last set contains the 2 remaining elements. Two players each choose a integer from 1 to 17 at random. The probability that they choose numbers from the same set can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. What is the value of \(a + b\)?

Details and assumptions

2 sets are disjoint if their intersection is the empty set.

The 5 disjoint sets contain 17 unique elements, hence each number only belongs in 1 set.

Since the players act independently of each other, and each chooses a number at random, they could end up choosing the same number.


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