A. how many numbers must be selected from the set1, 3, 5, 7, 9, 11, 13, 15 to guarantee that at least one pair of these numbers add up to 16?

We have been given a data set 1, 3, 5, 7, 9, 11, 13, 15 and we are asked how many numbers must be selected from the set to guarantee that at least one pair of these numbers add up to 16.

Now let us find such numbers from the data set who add up to 16. These numbers are (15,1), (7,9), (13,3) and (11,5).

We can see that 15 and 1 add up to 16, but we can not say that if we choose 2 numbers from our data they will add up to 16. Because if we choose last two numbers or starting two numbers of our given data set it will be false.

If we choose 3 numbers from our data it is not certain that at least one pair add up to 16. So is the case with choosing four numbers.

Now we have to choose such numbers which will certainly add up to 16.

We can see if we choose 5 numbers from our data at least one pair will add up to 16. Either these are initial 5 numbers from our data set or last 5 numbers of our data set.

Therefore, we can guarantee that if we choose 5 numbers from our data set at least one pair will add up to 16.