2. Micek is packing candies into bags. He has three types of candies: caramels, chocolate kisses, and gummy candies. If each bag must contain exactly 6 candies and at least 1 candy of each type, how many different bags can he make?
First method:

Let $k$ denote a caramel, $p$ denote a chocolate chip, and $g$ denote a gummy candy.

The problem can be solved by writing six-element sets of the form $\{k, p, g, -, -, -\}$.

The missing elements in the positions should be filled with the following subsets:

| $\{k, k, k\},\{p, p, p\},\{g, g, g\}$ | 1 POINT |
| :--- | ---: |
| $\{k, k, p\},\{k, k, g\}$ | 1 POINT |
| $\{p, p, k\},\{p, p, g\}$ | 1 POINT |
| $\{g, g, k\},\{g, g, p\}$ | 1 POINT |
| $\{k, p, g\}$ | 1 POINT |
| 10 different bags of candies can be composed. | 1 POINT |

Note: If the six-element sets (regardless of the order of the elements) are listed, they should be graded as suggested. For three six-element sets, 1 point; for five six-element sets, 2 points; for seven six-element sets, 3 points; for nine six-element sets, 4 points; for ten six-element sets, 5 points. The correct answer is worth 1 point. For repeated solutions, points are deducted, with 1 point deducted for every 2 repetitions, up to a maximum of 0 points.