combinaTION

D. This problem is best done by brute force. First, we must establish that Dan has the option
of taking one or two classes per week. If he only takes one class, the three possibilities are Y, W,
and K. If he takes two classes per week, the six possibilities are YY, KK, WW, YK, YW, and WK
(because the problem is looking for combinations, not arrangements). So that adds up to nine
possible combinations.

This question can be interpreted in different ways:

1. One way is that on one week night, only one class is offered
Selecting 1 class: 3 choices available so 3 ways (or 3C1 = 3)
Selecting 2 classes: Out of 3 classes, 2 need to be selected or we can say, 1 needs to be dropped. This can be done in 3 ways (or 3C2 = 3).
Total 6 ways.

2.i. The gym offers classes 3 nights a week (assume Mon, Wed, Fri). On ANY given class night, options of Yoga, WT or KB are available.
If we want to find the combination of classes only, not days, we can do it in the following way:
Selecting 1 class: 3 choices available so 3 ways (or 3C1 = 3)
Selecting 2 classes: Out of 3 classes, we need to choose 2 This can be done in 3 ways (or 3C2). Since we can choose the same class on both the days, we have another 3 ways.
Total 9 ways.

2.ii. If we want to find the combination of days and classes, we can do it in the following way:
Selecting 1 class: First we choose a day for which we have 3 options (or 3C1) and then we choose one of the 3 classes for which we again have 3 options (or 3C1) and hence we choose in 3x3 = 9 ways
Selecting 2 classes: First we choose 2 days out of 3 in 3 ways (or 3C2). Then we choose a class for the first day in 3 ways and choose a class for the second day in 3 ways. In all, we can choose 2 classes in 3x3x3 = 27 ways.
Total 36 ways.

Now the question is which method to use here. 2.ii. is anyway not possible since the answer options do not have 36. It would be more reasonable to go with 2.i. since the question says "On any given
class night, Dan has the option of taking yoga, weight training, or kickboxing classes".