- #1

Lilia

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**Mod note**: Moved from a technical math forum, so missing the homework template.

Given a set A = {a

_{1}, a

_{2}, ..., a

_{n}} and its two subsets - X, Y so that these subsets satisfy a condition, find the number of such possible (X, Y) pairs

Condition 1: X ∩ Y = {a

_{1}, a

_{2}},

Condition 2: X ⊆ Y,

Condition 3: | (X - Y) ∪ (Y - X) | = 1

I've tried to solve these but I still can't figure out the right way to think when solving these.

For X ∩ Y = ∅ the number of subsets is ΣC(n,k) × 2

^{n-k}, where k=0÷n.

This is how I think - so, for Condition 1, X and Y should have only a

_{1}and a

_{2}in common and can have other distinct elements, so we need to choose those two elements but how to choose them for both X and Y?

For Condition 2, |X| can be k, |Y| can be k (or k-1?), therefore the number of subsets - C(n,k) × C(n,k) = (C(n,k))

^{2}? Is this right?

For Condition 3, X is a subset of Y, or Y is a subset of X, and they have one element in common, right? If so, how to count the number of subsets?

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