**Exercises:**

- If , show that
- Prove by induction on n that has subsets.
- Show that
- Show that for each
- Show that
- Show that
- Show that
- Show that and use Stirling’s formula to prove that
- Give a solution using binomial coefficients and a direct combinatorial solution to the following question: How many pairs of subsets of are there such that ?
- Show that the number of even subsets of equals the number of odd subsets of . Give two proofs, one using binomial formula, and one using a direct bijection. Calculate the sum of the sizes of all even (odd) subsets of .

More later,

Nalin Pithwa