Pascal's Triangle is a triangular array of numbers built by starting with 1 at the top and placing each new number as the sum of the two numbers above it. It matters because the same simple pattern appears in algebra, counting, probability, and number theory. Students use it to expand powers like (a + b)^n and to count combinations without listing every possibility.
Its visual structure makes hidden patterns in mathematics easier to see.
Each row of Pascal's Triangle contains binomial coefficients, which count how many ways a certain number of items can be chosen from a larger set. The triangle is symmetric because choosing k items from n is equivalent to leaving out n - k items. Diagonals reveal counting sequences such as natural numbers, triangular numbers, and tetrahedral numbers.
When entries are colored by divisibility, the triangle also shows fractal-like patterns that connect arithmetic to geometry.
Key Facts
- Each entry is found by adding the two entries above it: C(n, k) = C(n - 1, k - 1) + C(n - 1, k).
- The outside edges of Pascal's Triangle are always 1 because C(n, 0) = 1 and C(n, n) = 1.
- Row n gives the coefficients of (a + b)^n, so (a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4.
- The sum of the entries in row n is 2^n.
- The triangle is symmetric: C(n, k) = C(n, n - k).
- The formula for an entry is C(n, k) = n! / (k!(n - k)!).
Vocabulary
- Pascal's Triangle
- A triangular arrangement of numbers where each interior number is the sum of the two numbers directly above it.
- Binomial coefficient
- A number C(n, k) that counts the ways to choose k objects from n objects and appears in binomial expansions.
- Combination
- A selection of items where order does not matter.
- Row
- A horizontal level of Pascal's Triangle, often numbered starting with row 0 at the top.
- Symmetry
- A matching pattern on opposite sides of a shape or arrangement, such as equal entries across the center of Pascal's Triangle.
Common Mistakes to Avoid
- Starting row numbers at 1 without checking the convention. Many formulas use row 0 at the top, so using row 1 can shift every answer by one row.
- Adding the wrong pair of numbers when building the triangle. Each interior entry must be the sum of the two entries directly above it, not the numbers beside it in the same row.
- Forgetting that Pascal's Triangle coefficients match descending and ascending powers. In (a + b)^n, the powers of a decrease from n to 0 while the powers of b increase from 0 to n.
- Treating combinations as if order matters. C(n, k) counts unordered selections, so choosing A then B is the same as choosing B then A.
Practice Questions
- 1 Write row 6 of Pascal's Triangle, using row 0 as the top row.
- 2 Use Pascal's Triangle to expand (x + 2)^5.
- 3 Explain why row 5 of Pascal's Triangle is symmetric without just saying that the numbers look the same.