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**1. The problem statement, all variables and given/known data**

##H=\{1,2,3,\dots , 10\}##. Determine the following number: ## \#([G,F];G\subseteq F \subseteq H)##.

**2. Relevant equations**

**3. The attempt at a solution**

Here is my reasoning. If ##G=\emptyset##, there are ##2^{10}## ways to choose ##F##, so ##\binom{10}{0}2^{10}## total. If ##|G|=1##, there are ##2^9## ways to choose ##F##, so ##\binom{10}{1}2^9## ways total. And so on, until ##|G|=10##, so there is only one way to choose ##F##, and so ##\binom{10}{10}\cdot 1## ways in total.

These are are disjoint cases, so we sum them up: $$\sum_{k=0}^{10}\binom{10}{10-k}2^{k} = \sum_{k=0}^{10}\binom{10}{k}2^{k} = 3^{10}$$.

Does this all look like the correct reasoning?