The binomial coefficient is derived from Pascal’s triangle, which is a fact I remember but cannot explain, based on some hazy memories of 10th grade math class.
The binomial coefficient is derived from Pascal’s triangle, which is a fact I remember but cannot explain, based on some hazy memories of 10th grade math class.
You can take the two variables to be outcomes of a binomial event:
- at zero events: we have one outcome, nothing
- at one event: we have two outcomes, 1X and 1Y
- at two events, each of those two outcomes have two outcomes, but one X and one Y outcome both become XY, so we have 1X^2 2XY 1Y^2.
You can continue combining this way, adding an X and Y to each term (for the two possible outcomes) — and then grouping like terms.