### Instruction: Suppose two distinct integers are chosen from between 5 and 17, inclusive.  What is the probability that their product is odd?### Response: I want to find the probability of an event, so I need to compare the number of favorable outcomes to the number of possible outcomes.
The possible outcomes are all the ways I can choose two integers from between 5 and 17, inclusive.
I can use the combination formula to find this: (n choose r) = n! / (r! (n-r)!)
In this case, n = 13, since there are 13 integers between 5 and 17, and r = 2, since I'm choosing two of them.
So the number of possible outcomes is (13 choose 2) = 13! / (2! 11!) = (13 * 12) / (2 * 1) = 78.
The favorable outcomes are those where the product of the two integers is odd.
I know that the product of two integers is odd only if both integers are odd.
So I need to count how many ways I can choose two odd integers from between 5 and 17, inclusive.
There are 7 odd integers in this range: 5, 7, 9, 11, 13, 15, and 17.
So I can use the same combination formula, but with n = 7 and r = 2.
The number of favorable outcomes is (7 choose 2) = 7! / (2! 5!) = (7 * 6) / (2 * 1) = 21.
The probability of the event is the ratio of favorable to possible outcomes.
So the probability is 21 / 78, which can be simplified by dividing both numerator and denominator by 3.
The final answer is 7 / 26.