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Discrete math Subsets Contained

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I have another discrete problem I need help on.

It says: How many subsets contain 1 or 2 or 3 in the set {1,2,...,20}?

So my teacher told me that {1} is a subset, {1,3,4,5,19} would be a subset (i just chose that randomly), {2} would be a subset, {2,5,6,7,20} would be a subset (again, i just chose that at random, and continue to do that for all possible cominations.

I'm just not sure how to find the combinations. I don't think the numbers can be repeated.

So, i'm thinking that say for the number 3,.. would you have 17! possible combinations?
Because you would have
But...then you would have sets like

uhg...i dont understand. If you guys don't either, no worries, this stuff is hard!

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The expert examines discrete math subsets contained.

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How many subsets contain 1 or 2 or 3 in the set {1,2,...,20}?
Subsets contain 1 element:
{1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}, {10}, {11}, {12}, {13}, {14}, {15}, {16}, {17}, ...

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