Expanding Powers

Combinatorics · Axiom Academy

Discover the hidden patterns in polynomial expansions. Expanding by hand seems impossible. But what if there's a pattern? Notice: Each expansion gets longer. Can you spot a pattern in the coefficients? Let's look closer at the coefficients in these expansions. Notice anything special? Use the slider to explore different powers of (x+y). Watch how the coefficients match Pascal's Triangle! The coefficients aren't random - they count something important! To get each term, we pick either x or y from each factor: x³: Pick x from all 3 factors → 1 way x²y: Pick x from 2 factors, y from 1 → 3 ways xy²: Pick x from 1 factor, y from 2 → 3 ways y³: Pick y from all 3 factors → 1 way Now you've discovered the pattern behind one of mathematics' most important formulas! where C(n,k) = n!/(k!(n-k)!) is the binomial coefficient Pascal's Triangle generates all binomial coefficients There are (n+1) terms in the expansion of (x+y) n Coefficients count combinations: C(n,k) ways to choose k from n Powers of x decrease while powers of y increase The pattern works for ANY values of x and y! You'll learn to compute binomial coefficients efficiently, prove the binomial theorem, and apply it to solve real problems in probability, algebra, and beyond!

This is the written version of the interactive lesson above. See the full Combinatorics course.