Modeling a power as sum of combination

[johnhaddad]

Homework Statement

I am trying to come up with a logical explanation (using an example from real life modeled in Combination and/or Permutation) to show that 3^n = summation (from i=0 to n) of (n choose i)*(2^i). In other words, I am trying to pose a combinatorial/selection/combination question that then can be modeled as both sides of the equation.

Homework Equations

Combinations and Permutations.

The Attempt at a Solution

My trial is that 3^n could be modeled as having 3 boxes and we are placing n items in it (eg. balls). So, if we pick n=5, it is as selecting 3 balls to put in box one, then returning them, then selecting 3 balls to put in box 2, then returning..etc while the order of selection is "don't care". However, I still can't show how that translates into the right side of the equation. Any help or pointers?

 

[NascentOxygen]

This is a half-baked idea, see whether you can make anything of it. I have not thought it through, though, not thoroughly. (Really, I haven't.)

You have discovered a strange mutant strain of yeast cell, here's a slide specimen I prepared ->

Each cell continuously grows in size (diameter) during the period of your study, and on a weekly basis each cell produces a pair of tiny but reproductively-mature offspring. This means the population triples in size every week. You happen to start with an initial population of just 3 cells.

With this scenario, you behold an ever-increasing range of sizes in your yeast culture, with the original 3 always largest in diameter, followed by their first daughters comprising the 6 next largest, followed by 18 same-sized cells (being a mix of daughters and grand-daughters of the original trio), etc.

After 12 weeks, determine how many of these cells populate your laboratory.