Two different numbers are selected simultaneously and at random from the set {1,2,3,4,5,6,7}. What is the probability that the positive difference between the two numbers is 3 or greater? Express your answer as a common fraction.
+1223 There are a total of \({7 \choose 2} = 21\) different pairs. We will count the pairs where the difference is 1 or 2. Note that a difference of 0 is not possible since the numbers have to be distinct.
Case 1: difference = 1
7-6, 6-5, 5-4, 4-3, 3-2, 2-1
6 pairs
Case 2: difference = 2
7-5, 6-4, 5-3, 4-2, 3-1
5 pairs
\(\text{probability} = 1 - \frac{11}{21} = \boxed{\frac{10}{21}}\)
