Binominal Theorem

Question

Could anyone help me with homework or give me a hint? Any help would be highly appreciated.

Given a set of N distinct objects:

How many ways are there to pick any number of them to be in a pile while the rest are in anotherpile? If your answer is written in terms of binomial coecients, use the Binomial Theorem to write as a single (N-dependent) number.

Thanks Daniel

Answer 1

Think about going to each object and flipping a switch on it, L or R, to decide which pile it goes in. How many choices do you have to make in this process? Now make sure to divide by two because we don't care to distinguish between the left and the right piles.

Answer 2

Hint

1. There are the following choices for the number of objects in the first pile: 0, 1, ..., N
2. If the first pile has $k$, how much does the second pile have?
3. To reach such an arrangement, pick $k$ elements for the first pile from the entire group. Is order important? How many ways are there to do that (this involves a binomial coefficient)?
4. Add all the ways from step (3) -- for different values of $k$ -- and use Binomial Theorem.

Answer 3

Hint 1: There are $\binom{n}{k}$ ways to put $k$ items into the first pile and $n-k$ items into the second pile. Consider what this means for all $k$.

Hint 2: Each item can either be in the first pile or the second pile ($2$ options for each item).