Key idea. Make pairs whose sums are 11.

1. Arrange the available numbers in the five pairs

   \[(1,10),\quad(2,9),\quad(3,8),\quad(4,7),\quad(5,6).\]

2. Each number from 1 to 10 belongs to exactly one of these pairs.

3. If we chose at most one number from each pair, we could choose at most five numbers. Choosing six distinct numbers therefore forces both members of some pair.

4. The two members of that pair have sum 11, as required.

Remember. The useful “boxes” may be pairs that you construct, rather than groups already named in the question.