suppose 2 distinct integers are from 5 to 17. what is the probability that their product is odd?

There are 7 odd numbers and 6 even numbers in the range.  To get an odd product both of the chosen random numbers must be odd.  

probability that the first number is odd: p1 = 7/13

probability that the second number is odd: p2 = 6/12 → 1/2  (I assume the word "distinct" means the first number is removed from the set before choosing the second number).

Overall probability:   p1*p2 = 7/13*1/2 → 7/26 ≈ 0.269

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Alan's answer is correct......Another way to see this is that there are 13 numbers beween 5 and 17, inclusive........the number of possible sets formed by choosing an two of these  = C(13,2)

However, we are only interesed in the sets formed by choosing any two of the odd numbers [ since their product is also odd]......and this = C(7,2)  ...so....the probability  =

C(7,2)  / C(13,2)  ≈  0.269