Suppose two distinct integers are chosen from between 5 and 17, inclusive.  What is the probability that their product is odd?
There are 13 integers between 5 and 17 inclusive, so there are $\binom{13}{2} = 78$ ways to choose two of them without regard to order.  In order for the product of two integers to be odd, both of the integers themselves must be odd. There are 7 odd integers between 5 and 17 inclusive, so there are $\binom72 = 21$ ways to choose two of them without regard to order.  Therefore, the desired probability is $\dfrac{21}{78} = \boxed{\dfrac{7}{26}}$.
\dfrac{7}{26}
Level 3
Counting & Probability
Problem: Suppose two distinct integers are chosen from between 5 and 17, inclusive.  What is the probability that their product is odd?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the probability that the product of two distinct integers chosen from the range 5 to 17 is odd, we first need to identify the total number of integers in this range. The integers from 5 to 17 are: 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17. This gives us a total of 13 integers.

Next, we need to understand the conditions under which the product of two integers is odd. A product is odd if and only if both integers are odd. Therefore, we must count the odd integers within the specified range.

The odd integers between 5 and 17 are: 5, 7, 9, 11, 13, 15, 17. There are 7 odd integers in total.

Now, we calculate the total number of ways to choose 2 distinct integers from the 13 available integers. This can be calculated using the combination formula C(n, k), where n is the total number of items to choose from, and k is the number of items to choose. Thus, the total combinations of choosing 2 integers from 13 is:

C(13, 2) = 13! / (2!(13-2)!) = (13 * 12) / (2 * 1) = 78.

Next, we calculate the number of ways to choose 2 odd integers from the 7 odd integers. Using the same combination formula, we find:

C(7, 2) = 7! / (2!(7-2)!) = (7 * 6) / (2 * 1) = 21.

Now, we can find the probability that the product of the two chosen integers is odd. This probability is the ratio of the number of favorable outcomes (choosing 2 odd integers) to the total outcomes (choosing any 2 integers):

Probability = Number of ways to choose 2 odd integers / Total ways to choose 2 integers = 21 / 78.

To simplify this fraction, we find the greatest common divisor (GCD) of 21 and 78, which is 3. Dividing both the numerator and the denominator by 3 gives us:

21 / 3 = 7 and 78 / 3 = 26.

Thus, the simplified probability that the product of the two distinct integers is odd is:

7 / 26.<|END_OF_LATENT|> The final answer is $\dfrac{7}{26}$.