Pigeonhole Principle - Show two subsets have the same age
Solution 1:
The sum of ages cannot exceed $100 \times 10=1000$ $(1001$ cages, including $0 )$. The total ways to chose subsets of people are $2^{10}=1024$ $(1024$ pigeons$)$.
Can you see that coming?
Solution 2:
Hint: how many subsets are there? What is the maximum total age of a subset?