Binomial coefficients, subsets, and the binomial theorem
While permutations care about order, combinations count unordered selections. "How many ways can you choose k items from n?" is answered by the binomial coefficient C(n, k) = n! / (k!(n-k)!). These numbers form Pascal's triangle, one of the most pattern-rich objects in all of mathematics.
In this lesson, you will visualize subset selection, explore the hidden patterns in Pascal's triangle, and see how binomial coefficients power the expansion of (a+b)ⁿ.
The binomial coefficient C(n, k) counts the number of ways to choose k items from a set of n, where order does not matter. Notice the symmetry: C(n, k) = C(n, n-k), because choosing which k to include is equivalent to choosing which n-k to exclude.
Filled circles indicate selected items. Each row is one of the C(n,k) possible subsets.
Key insight: C(n, k) = C(n, n-k) is not just an algebraic identity -- it reflects a bijection: every k-element subset corresponds to a unique (n-k)-element complement.
Pascal's triangle arranges binomial coefficients in rows: row n contains C(n, 0), C(n, 1), ..., C(n, n). Each entry equals the sum of the two entries above it. Toggle highlighting modes to discover hidden patterns: the Sierpinski triangle from odd/even coloring, modular arithmetic patterns, and diagonal sequences.
Each entry C(n,k) equals the sum of the two entries above it. Toggle highlight modes to reveal hidden patterns.
Key insight: Coloring Pascal's triangle by parity (odd = colored, even = dark) reveals the Sierpinski triangle fractal. This happens because C(n, k) mod 2 = 1 if and only if every bit of k is also a bit of n (Lucas' theorem).
The binomial theorem states that (a + b)ⁿ = Σ C(n, k) · aⁿ⁻ᵏ · bᵏ. Each term corresponds to choosing which k of the n factors contribute b (and the rest contribute a). Watch the expansion unfold term by term, with each coefficient linked to its position in Pascal's triangle.
Click any term to highlight it and its Pascal's triangle entry. Use auto-play to step through each term.
Key insight: The binomial theorem is not just a formula -- it is a counting statement. The coefficient C(n, k) counts the number of ways to pick k copies of b from n factors of (a + b).