Answer:
a: discrete
b: discrete
c: continuous
d: discrete
Explanation:
A discrete set is a set with countable elements, while a continuous set is a dense set.
a) The number of buttons in a box.
Suppose that there are 30 buttons in a box.
We only can draw 1 button, 2 buttons, 3 buttons, etc.
So there are a lot of values that we can not get (like 1.5, 3.498, etc)
So this is a discrete set.
b) The weight of sweets in a bag.
Suppose that each sweet weights W.
Then the weight of the bag will be:
weight = B + n*W
Where B is the weight of the bag itself, and n is the number of sweets.
Now, suppose that we add another sweet, then we get:
weight = B + (n + 1)*W
Then we can see that this is not a dense set. The weight can increase (or decrease) only on multiples of W. Then this is a discrete set.
c) The volume of liquid in a bath.
Here we can add any volume of liquid that we want to the bath (it only would be discrete if we saw the system in a microscopical way) so the volume of liquid in the bath can be any value between zero and the maximum volume of liquid that the bath can have, then this is a continuous set.
d) The number of children at a swimming pool.
Similar to the case of the number of buttons in a box, here we only can have whole numbers, then this is a discrete set.