Eeny, meeny, miny, moe
The integers 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 , where and are coprime positive integers. What is the value of ?
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.
The answer is 352.
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1 solution
For each set, we calculate the probability that both players chose a number from that set. For the set of size 5, this probability is ( 1 7 5 ) 2 = 2 8 9 2 5 . For the set of size 4, it is 2 8 9 1 6 . For each set of size 3 it is 2 8 9 9 and for the set of size 2 it is 2 8 9 4 . So the probability of them being in the same set is 2 8 9 2 5 + 1 6 + 9 + 9 + 4 = 2 8 9 6 3 . So a + b = 6 3 + 2 8 9 = 3 5 2 .